Displaying 1-10 of 11 results found.
Completely multiplicative with a(prime(k)) = prime(k+1).
+10
873
1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23, 63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, 37, 243, 65, 57, 77, 225, 41, 69, 85, 189, 43, 165, 47, 117, 175, 87, 53, 405, 121, 147, 95, 153, 59, 375, 91, 297, 115, 93, 61, 315, 67, 111, 275, 729, 119
COMMENTS
Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021 Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021 a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001 Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.
(End)
REFERENCES
Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).
FORMULA
If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 2.06399637... . - Amiram Eldar, Nov 18 2022
EXAMPLE
a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.
MAPLE
a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
MATHEMATICA
a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ (a[#1]^#2& @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Dec 01 2011, updated Sep 20 2019 *) Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1], {n, 65}] (* Michael De Vlieger, Mar 24 2017 *)
PROG
(PARI) a(n)=local(f); if(n<1, 0, f=factor(n); prod(k=1, matsize(f)[1], nextprime(1+f[k, 1])^f[k, 2]))
(PARI) a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Michel Marcus, May 17 2014 (Haskell)
a003961 1 = 1
a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(Perl) use ntheory ":all"; sub a003961 { vecprod(map { next_prime($_) } factor(shift)); } # Dana Jacobsen, Mar 06 2016 (Python)
from sympy import factorint, prime, primepi, prod
def a(n):
f=factorint(n)
return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)
CROSSREFS
Cf. A064989 (a left inverse), A064216, A000040, A002110, A000265, A027746, A046523, A048673 (= (a(n)+1)/2), A108228 (= (a(n)-1)/2), A191002 (= a(n)*n), A252748 (= a(n)-2n), A286385 (= a(n)-sigma(n)), A283980 (= a(n)* A006519(n)), A341529 (= a(n)*sigma(n)), A326042, A049084, A001221, A001222, A122111, A225546, A260443, A245606, A244319, A246269 (= A065338(a(n))), A322361 (= gcd(n, a(n))), A305293. Cf. also following permutations and other sequences that can be defined with the help of this sequence: A005940, A163511, A122111, A260443, A206296, A265408, A265750, A275733, A275735, A297845, A091202 & A091203, A250245 & A250246, A302023 & A302024, A302025 & A302026. Applying the same transformation again gives A357852.
Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).
+10
96
1, 2, 4, 3, 16, 8, 256, 6, 9, 32, 65536, 12, 4294967296, 512, 64, 5, 18446744073709551616, 18, 340282366920938463463374607431768211456, 48, 1024, 131072, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 24, 81, 8589934592, 36, 768
COMMENTS
This is a multiplicative self-inverse permutation of the integers.
This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms. This sequence provides a relationship between the operations of squaring and prime shift ( A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row. A329050 gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.
Alternative construction: For n > 1, form a vector v of length A299090(n), where each element v[i] for i=1.. A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1.. A299090(n)} A000040(i)^ A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n). This permutation effects the following mappings:
(End)
Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors). (End)
FORMULA
a(prime(i)) = 2^(2^(i-1)).
The previous formula implies a(n*k) = a(n) * a(k) if A059895(n,k) = 1. (End)
For all n >= 1, a(2n) = A334747(a(n)). In particular, for n = A003159(m), m >= 1, a(2n) = 2*a(n). [Note that A003159 includes all odd numbers] (End)
EXAMPLE
7744 = prime(1)^2^(2-1)*prime(1)^2^(3-1)*prime(5)^2^(2-1).
a(7744) = prime(2)^2^(1-1)*prime(3)^2^(1-1)*prime(2)^2^(5-1) = 645700815.
MATHEMATICA
Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 28] (* Michael De Vlieger, Jan 21 2020 *)
PROG
(PARI)
A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
a(n) = {my(f=factor(n)); for (i=1, #f~, my(p=f[i, 1]); f[i, 1] = A019565(f[i, 2]); f[i, 2] = 2^(primepi(p)-1); ); factorback(f); } \\ Michel Marcus, Nov 29 2019 (PARI)
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A225546(n) = if(1==n, 1, my(f=factor(n), u=#binary(vecmax(f[, 2])), prods=vector(u, x, 1), m=1, e); for(i=1, u, for(k=1, #f~, if(bitand(f[k, 2], m), prods[i] *= f[k, 1])); m<<=1); prod(i=1, u, prime(i)^ A048675(prods[i]))); \\ Antti Karttunen, Feb 02 2020
(Python)
from math import prod
from sympy import prime, primepi, factorint
def A225546(n): return prod(prod(prime(i) for i, v in enumerate(bin(e)[:1:-1], 1) if v == '1')**(1<<primepi(p)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 17 2023
CROSSREFS
Cf. A225547 (fixed points) and the subsequences listed there. Pairs of sequences (f,g) that satisfy a(f(n)) = g(a(n)): ( A000265, A008833), ( A000290, A003961), ( A005843, A334747), ( A006519, A007913), ( A008586, A334748). Pairs of sequences (f,g) that satisfy f(a(n)) = g(n), possibly with offset change: ( A000035, A010052), ( A008966, A209229), ( A007814, A248663), ( A061395, A299090), ( A087207, A267116), ( A225569, A227291). Cf. A331740 [= A001222(a(n)), number of prime factors with multiplicity]. A self-inverse isomorphism between pairs of A059897 subgroups: ( A000079, A005117), ( A000244, A062503), ( A000290\{0}, A005408), ( A000302, A056911), ( A000351, A113849 U {1}), ( A000400, A062838), ( A001651, A252895), ( A003586, A046100), ( A007310, A000583), ( A011557, A113850 U {1}), ( A028982, A042968), ( A053165, A065331), ( A262675, A268390).
Encoded multiplication table for polynomials in one indeterminate with nonnegative integer coefficients. Symmetric square array T(n, k) read by antidiagonals, n > 0 and k > 0. See comment for details.
+10
24
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 7, 16, 7, 6, 1, 1, 7, 15, 25, 25, 15, 7, 1, 1, 8, 11, 36, 11, 36, 11, 8, 1, 1, 9, 27, 49, 35, 35, 49, 27, 9, 1, 1, 10, 25, 64, 13, 90, 13, 64, 25, 10, 1, 1, 11, 21, 81, 125, 77, 77, 125, 81
COMMENTS
For any number n > 0, let f(n) be the polynomial in a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials in a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).
This table has many similarities with A248601. For any n > 0 and m > 0, f(n * m) = f(n) + f(m).
Also, f(1) = 0 and f(2) = 1.
The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.
See A297473 for the main diagonal of T. As a binary operation, T(.,.) is related to A306697(.,.) and A329329(.,.). When their operands are terms of A050376 (sometimes called Fermi-Dirac primes) the three operations give the same result. However the rest of the multiplication table for T(.,.) can be derived from these results because T(.,.) distributes over integer multiplication ( A003991), whereas for A306697 and A329329, the equivalent derivation uses distribution over A059896(.,.) and A059897(.,.) respectively. - Peter Munn, Mar 25 2020 The operation defined by this sequence can be extended to be the multiplicative operator of a ring over the positive rationals that is isomorphic to the polynomial ring Z[x]. The extended function f (described in the author's original comments) is the isomorphism we use, and it has the same relationship with the extended operation that exists between their unextended equivalents.
Denoting this extension of T(.,.) as t_Q(.,.), we get t_Q(n, 1/k) = t_Q(1/n, k) = 1/T(n, k) and t_Q(1/n, 1/k) = T(n, k) for positive integers n and k. The result for other rationals is derived from the distributive property: t_Q(q, r*s) = t_Q(q, r) * t_Q(q, s); t_Q(q*r, s) = t_Q(q, s) * t_Q(r, s). This may look unusual because standard multiplication of rational numbers takes on the role of the ring's additive group.
There are many OEIS sequences that can be shown to be a list of the integers in an ideal of this ring. See the cross-references.
There are some completely additive sequences that similarly define by extension completely additive functions on the positive rationals that can be shown to be homomorphisms from this ring onto the integer ring Z, and these functions relate to some of the ideals. For example, the extended function of A048675, denoted A048675_Q, maps i/j to A048675(i) - A048675(j) for positive integers i and j. For any positive integer k, the set {r rational > 0 : k divides A048675_Q(r)} forms an ideal of the ring; for k=2 and k=3 the integers in this ideal are listed in A003159 and A332820 respectively. (End)
LINKS
Eric Weisstein's World of Mathematics, Ring.
FORMULA
T is completely multiplicative in both parameters:
- for any n > 0
- and k > 0 with prime factorization Prod_{i > 0} prime(i)^e_i:
- T(prime(n), k) = T(k, prime(n)) = Prod_{i > 0} prime(n + i - 1)^e_i.
For any m > 0, n > 0 and k > 0:
- T(n, k) = T(k, n) (T is commutative),
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(n, 1) = 1 (1 is an absorbing element for T),
- T(n, 2) = n (2 is an identity element for T),
- T(n, 2^i) = n^i for any i >= 0,
- T(n, 3^i) = A003961(n)^i for any i >= 0, From Peter Munn, Mar 13 2020 and Apr 20 2021: (Start) T(n, m*k) = T(n, m) * T(n, k); T(n*m, k) = T(n, k) * T(m, k) (T distributes over multiplication).
(End)
EXAMPLE
Array T(n, k) begins:
n\k| 1 2 3 4 5 6 7 8 9 10
---+------------------------------------------------
3| 1 3 5 9 7 15 11 27 25 21 -> A003961 4| 1 4 9 16 25 36 49 64 81 100 -> A000290 5| 1 5 7 25 11 35 13 125 49 55 -> A357852 6| 1 6 15 36 35 90 77 216 225 210 -> A191002 7| 1 7 11 49 13 77 17 343 121 91
8| 1 8 27 64 125 216 343 512 729 1000 -> A000578 9| 1 9 25 81 49 225 121 729 625 441
10| 1 10 21 100 55 210 91 1000 441 550
The encoding, n, of polynomials, f(n), that is used for the table is further described in A206284. Examples of encoded polynomials: n f(n) n f(n)
1 0 16 4
2 1 17 x^6
3 x 21 x^3 + x
4 2 25 2x^2
5 x^2 27 3x
6 x + 1 35 x^3 + x^2
7 x^3 36 2x + 2
8 3 49 2x^3
9 2x 55 x^4 + x^2
10 x^2 + 1 64 6
11 x^4 77 x^4 + x^3
12 x + 2 81 4x
13 x^5 90 x^2 + 2x + 1
15 x^2 + x 91 x^5 + x^3
(End)
PROG
(PARI) T(n, k) = my (f=factor(n), p=apply(primepi, f[, 1]~), g=factor(k), q=apply(primepi, g[, 1]~)); prod (i=1, #p, prod(j=1, #q, prime(p[i]+q[j]-1)^(f[i, 2]*g[j, 2])))
CROSSREFS
Integers in the ideal of the related ring (see Jun 2021 comment) generated by S: S={3}: A005408, S={4}: A000290\{0}, S={4,3}: A003159, S={5}: A007310, S={5,4}: A339690, S={6}: A325698, S={6,4}: A028260, S={7}: A007775, S={8}: A000578\{0}, S={8,3}: A191257, S={8,6}: A332820, S={9}: A016754, S={10,4}: A340784, S={11}: A008364, S={12,8}: A145784, S={13}: A008365, S={15,4}: A345452, S={15,9}: A046337, S={16}: A000583\{0}, S={17}: A008366. Equivalent sequence for polynomial composition: A326376.
Table of powers of squarefree numbers, powers of A019565(n) in increasing order in row n. Square array A(n,k) n >= 0, k >= 0 read by descending antidiagonals.
+10
24
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 9, 6, 1, 1, 16, 27, 36, 5, 1, 1, 32, 81, 216, 25, 10, 1, 1, 64, 243, 1296, 125, 100, 15, 1, 1, 128, 729, 7776, 625, 1000, 225, 30, 1, 1, 256, 2187, 46656, 3125, 10000, 3375, 900, 7, 1, 1, 512, 6561, 279936, 15625, 100000, 50625, 27000, 49, 14
COMMENTS
Transposition of this table, that is reflection about its main diagonal, has subtle symmetries. For example, consider the unique factorization of a number into powers of distinct primes. This can be restated as factorization into numbers from rows 2^n (n >= 0) with no more than one from each row. Reflecting about the main diagonal, this factorization becomes factorization (of a related number) into numbers from columns 2^k (k >= 0) with no more than one from each column. This is also unique and is factorization into powers of squarefree numbers with distinct exponents that are powers of two. See the example section.
FORMULA
A(n,2k) = A000290(A(n,k)) = A(n,k)^2. A(n,2k+1) = A(n,2k) * A(n,1).
A(2n+1,k) = A(2n,k) * A(1,k).
A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)), = A329329(A(n,1), A(1,k)). Sum_{n>=0} 1/A(n,k) = zeta(k)/zeta(2*k), for k >= 2. - Amiram Eldar, Dec 03 2022
EXAMPLE
Square array A(n,k) begins:
n\k | 0 1 2 3 4 5 6 7
----+------------------------------------------------------------------
0| 1 1 1 1 1 1 1 1
1| 1 2 4 8 16 32 64 128
2| 1 3 9 27 81 243 729 2187
3| 1 6 36 216 1296 7776 46656 279936
4| 1 5 25 125 625 3125 15625 78125
5| 1 10 100 1000 10000 100000 1000000 10000000
6| 1 15 225 3375 50625 759375 11390625 170859375
7| 1 30 900 27000 810000 24300000 729000000 21870000000
8| 1 7 49 343 2401 16807 117649 823543
9| 1 14 196 2744 38416 537824 7529536 105413504
10| 1 21 441 9261 194481 4084101 85766121 1801088541
11| 1 42 1764 74088 3111696 130691232 5489031744 230539333248
12| 1 35 1225 42875 1500625 52521875 1838265625 64339296875
Reflection of factorization about the main diagonal: (Start)
The canonical (prime power) factorization of 864 is 2^5 * 3^3 = 32 * 27. Reflecting the factors about the main diagonal of the table gives us 10 * 36 = 10^1 * 6^2 = 360. This is the unique factorization of 360 into powers of squarefree numbers with distinct exponents that are powers of two.
Reflection about the main diagonal is given by the self-inverse function A225546(.). Clearly, all positive integers are in the domain of A225546, whether or not they appear in the table. It is valid to start from 360, observe that A225546(360) = 864, then use 864 to derive 360's factorization into appropriate powers of squarefree numbers as above. (End)
CROSSREFS
Rows (abbreviated list): A000079(1), A000244(2), A000400(3), A000351(4), A011557(5), A001024(6), A009974(7), A000420(8), A001023(9), A009965(10), A001020(16), A001022(32), A001026(64).
Square array A(n,k) = prime(n+1)^(2^k), read by descending antidiagonals (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...; Fermi-Dirac primes ( A050376) in matrix form, sorted into rows by their prime divisor.
+10
19
2, 4, 3, 16, 9, 5, 256, 81, 25, 7, 65536, 6561, 625, 49, 11, 4294967296, 43046721, 390625, 2401, 121, 13, 18446744073709551616, 1853020188851841, 152587890625, 5764801, 14641, 169, 17, 340282366920938463463374607431768211456, 3433683820292512484657849089281, 23283064365386962890625, 33232930569601, 214358881, 28561, 289, 19
COMMENTS
This sequence is a permutation of A050376, so every positive integer is the product of a unique subset, S_factors, of its terms. If we restrict S_factors to be chosen from a subset, S_0, consisting of numbers from specified rows and/or columns of this array, there are notable sequences among those that may be generated. See the examples. Other notable sequences can be generated if we restrict the intersection of S_factors with specific rows/columns to have even cardinality. In any of the foregoing cases, the numbers in the resulting sequence form a group under the binary operation A059897(.,.). Shares with array A246278 the property that columns grow downward by iterating A003961, and indeed, this array can be obtained from A246278 by selecting its columns 1, 2, 8, 128, ..., 2^((2^k)-1), for k >= 0. A(n,k) is the image of the lattice point with coordinates X=n and Y=k under the inverse of the bijection f defined in the first comment of A306697. This geometric relationship can be used to construct an isomorphism from the polynomial ring GF(2)[x,y] to a ring over the positive integers, using methods similar to those for constructing A297845 and A306697. See A329329, the ring's multiplicative operator, for details.
FORMULA
A(0,k) = 2^(2^k), and for n > 0, A(n,k) = A003961(A(n-1,k)). A(n,k+1) = A000290(A(n,k)) = A(n,k)^2.
EXAMPLE
The top left 5 X 5 corner of the array:
n\k | 0 1 2 3 4
----+-------------------------------------------------------
0 | 2, 4, 16, 256, 65536, ...
1 | 3, 9, 81, 6561, 43046721, ...
2 | 5, 25, 625, 390625, 152587890625, ...
3 | 7, 49, 2401, 5764801, 33232930569601, ...
4 | 11, 121, 14641, 214358881, 45949729863572161, ...
Column 0 continues as a list of primes, column 1 as a list of their squares, column 2 as a list of their 4th powers, and so on.
Every nonnegative power of 2 ( A000079) is a product of a unique subset of numbers from row 0; every squarefree number ( A005117) is a product of a unique subset of numbers from column 0. Likewise other rows and columns generate the sets of numbers from sequences: Column 1: A062503 Squares of squarefree numbers. Column 2: A113849 4th powers of squarefree numbers. Union of rows 0 and 1: A003586 3-smooth numbers. Union of columns 0 and 1: A046100 Biquadratefree numbers. Union of row 0 / column 0: A122132 Oddly squarefree numbers. Row 0 excluding column 0: A000302 Powers of 4. Column 0 excluding row 0: A056911 Squarefree odd numbers. All rows except 0: A005408 Odd numbers. All columns except 0: A000290\{0} Positive squares. All rows except 1: A001651 Numbers not divisible by 3. All columns except 1: A252895 (have odd number of square divisors). If, instead of restrictions on choosing individual factors of the product, we restrict the product to be of an even number of terms from each row of the array, we get A262675. The equivalent restriction applied to columns gives us A268390; applied only to column 0, we get A028260 (product of an even number of primes).
MATHEMATICA
Table[Prime[#]^(2^k) &[m - k + 1], {m, 0, 7}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Dec 28 2019 *)
PROG
(PARI)
up_to = 105;
A329050sq(n, k) = (prime(1+n)^(2^k));
A329050list(up_to) = { my(v = vector(up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A329050sq(col, a-col))); (v); };
v329050 = A329050list(up_to);
CROSSREFS
See also the table in the example section.
EXTENSIONS
Example annotated for clarity by Peter Munn, Feb 12 2020
Multiplicative operator of a ring over the positive integers that has A059897(.,.) as additive operator and is isomorphic to GF(2)[x,y] with A329050(i,j) the image of x^i * y^j.
+10
17
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 7, 16, 7, 6, 1, 1, 7, 15, 25, 25, 15, 7, 1, 1, 8, 11, 36, 11, 36, 11, 8, 1, 1, 9, 27, 49, 35, 35, 49, 27, 9, 1, 1, 10, 25, 64, 13, 10, 13, 64, 25, 10, 1, 1, 11, 21, 81, 125, 77, 77, 125, 81
COMMENTS
Square array A(n,k), n >= 1, k >= 1, read by descending antidiagonals.
The group defined by the binary operation A059897(.,.) over the positive integers is commutative with all elements self-inverse, and isomorphic to the additive group of GF(2) polynomial rings such as GF(2)[x,y]. There is a unique isomorphism extending each bijective mapping between respective minimal generating sets. The lexicographically earliest minimal generating set for the A059897 group is A050376, often called the Fermi-Dirac primes. This set has a natural arrangement in a square array, given by A329050(i,j) = prime(i+1)^(2^j), i >= 0, j >= 0. The most meaningful generating set for the additive group of GF(2)[x,y] is {x^i * y^j: i >= 0, j >= 0}, which similarly forms a square array. All this makes A329050(i,j) especially appropriate to be the image (under an isomorphism) of the GF(2) polynomial x^i * y^j. Using g to denote the intended isomorphism, we specify g(x^i * y^j) = A329050(i,j). This maps minimal generating sets of the additive groups, so the definition of g is completed by specifying g(a+b) = A059897(g(a), g(b)). We then calculate the image under g of polynomial multiplication in GF(2)[x,y], giving us this sequence as the matching multiplicative operator for an isomorphic ring over the positive integers. Using f to denote the inverse of g, A[n,k] = g(f(n) * f(k)). See the formula section for an alternative definition based on the A329050 array, independent of GF(2)[x,y]. Closely related to A306697 and A297845. If A059897 is replaced in the alternative definition by A059896 (and the definition is supplemented by the derived identity for the absorbing element, shown in the formula section), we get A306697; if A059897 is similarly replaced by A003991 (integer multiplication), we get A297845. This sequence and A306697, considered as multiplicative operators, are carryless arithmetic equivalents of A297845. A306697 uses a method analogous to binary-OR when there would be a multiplicative carry, while this sequence uses a method analogous to binary exclusive-OR. In consequence A(n,k) <> A297845(n,k) exactly when A306697(n,k) <> A297845(n,k). This relationship is not symmetric between the 3 sequences: there are n and k such that A(n,k) = A306697(n,k) <> A297845(n,k). For example A(54,72) = A306697(54,72) = 273375000 <> A297845(54,72) = 22143375000.
LINKS
Eric Weisstein's World of Mathematics, Ring
FORMULA
Alternative definition: (Start)
(End)
Derived identities: (Start)
A(n,1) = A(1,n) = 1 (1 is an absorbing element).
A(n,2) = A(2,n) = n.
A(n,k) = A(k,n).
A(n, A(m,k)) = A(A(n,m), k).
(End)
EXAMPLE
Square array A(n, k) begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12
---+-------------------------------------------------------------
1| 1 1 1 1 1 1 1 1 1 1 1 1
2| 1 2 3 4 5 6 7 8 9 10 11 12
3| 1 3 5 9 7 15 11 27 25 21 13 45
4| 1 4 9 16 25 36 49 64 81 100 121 144
5| 1 5 7 25 11 35 13 125 49 55 17 175
6| 1 6 15 36 35 10 77 216 225 210 143 540
7| 1 7 11 49 13 77 17 343 121 91 19 539
8| 1 8 27 64 125 216 343 32 729 1000 1331 1728
9| 1 9 25 81 49 225 121 729 625 441 169 2025
10| 1 10 21 100 55 210 91 1000 441 22 187 2100
11| 1 11 13 121 17 143 19 1331 169 187 23 1573
12| 1 12 45 144 175 540 539 1728 2025 2100 1573 80
PROG
(PARI) \\ See Links section.
a(n) = Product_{k=1..n} prime(k)^(2^(n-k)).
+10
12
1, 2, 12, 720, 3628800, 144850083840000, 272760108249915378892800000000, 1264767303092594444142256488682840323816161280000000000000000
COMMENTS
x^(2^n) - a(n) is the minimal polynomial over Q for the algebraic number sqrt(p(1)*sqrt(p(2)*...*sqrt(p(n-1)*sqrt(p(n)))...)), where p(k) is the k-th prime. Each such monic polynomial is irreducible by Eisenstein's Criterion (using p = p(n)).
A prime version of Somos's quadratic recurrence sequence A052129(n) = A052129(n-1)^2 * n = Product_{k=1..n} k^(2^(n-k)). - Jonathan Sondow, Mar 29 2014 All positive integers have unique factorizations into powers of distinct primes, and into powers of squarefree numbers with distinct exponents that are powers of 2. (See A329332 for a description of the relationship between the two.) a(n) is the least number such that both factorizations have n factors. - Peter Munn, Dec 15 2019 From Peter Munn, Jan 24 2020 to Feb 06 2020: (Start) For n >= 0, a(n+1) is the n-th power of 12 in the monoid defined by A306697. a(n) is the least positive integer that cannot be expressed as the product of fewer than n terms of A072774 (powers of squarefree numbers). All terms that are less than the order of the Monster simple group ( A003131) are divisors of the group's order, with a(6) exceeding its square root. (End)
It is remarkable that 4 of the first 5 terms are factorials. - Hal M. Switkay, Jan 21 2025
FORMULA
For n > 0, a(n) = a(n-1)^2 * prime(n); a(0) = 1. [edited to extend to a(0) by Peter Munn, Feb 13 2020] a(0) = 1; for n > 0, a(n) = 2^(2^(n-1)) * A003961(a(n-1)). - Antti Karttunen, Feb 06 2016, edited Feb 13 2020 because of the new prepended starting term.
EXAMPLE
a(1) = 2^1 = 2 and x^2 - 2 is the minimal polynomial for the algebraic number sqrt(2).
a(4) = 2^8*3^4*5^2*7^1 = 3628800 and x^16 - 3628800 is the minimal polynomial for the algebraic number sqrt(2*sqrt(3*sqrt(5*sqrt(7)))).
MAPLE
a:= proc(n) option remember;
`if`(n=0, 1, a(n-1)^2*ithprime(n))
end:
MATHEMATICA
RecurrenceTable[{a[1] == 2, a[n] == a[n-1]^2 Prime[n]}, a, {n, 10}] (* Vincenzo Librandi, Feb 06 2016 *) Table[Product[Prime[k]^2^(n-k), {k, n}], {n, 0, 10}] (* or *) nxt[{n_, a_}]:={n+1, a^2 Prime[n+1]}; NestList[nxt, {0, 1}, 10][[All, 2]] (* Harvey P. Dale, Jan 07 2022 *)
PROG
(PARI) a(n) = prod(k=1, n, prime(k)^(2^(n-k)))
(Scheme)
;; Two variants, both with memoization-macro definec.
(Magma) [n le 1 select 2 else Self(n-1)^2*NthPrime(n): n in [1..10]]; // Vincenzo Librandi, Feb 06 2016
1, 2, 9, 12, 18, 24, 80, 108, 160, 216, 625, 720, 960, 1250, 1440, 1792, 1920, 2025, 3584, 4050, 5625, 7500, 8640, 11250, 15000, 16128, 17280, 18225, 21504, 24300, 32256, 36450, 43008, 48600, 50000, 67500, 100000, 135000, 143360, 162000, 193536, 218700, 286720, 321489, 324000, 387072, 437400, 450000, 600000
COMMENTS
Every number in this sequence is the product of a unique subset of A225548. The terms are the numbers whose Fermi-Dirac factors (see A050376) occur symmetrically about the main diagonal of A329050. Closed under the commutative binary operation A059897(.,.). As numbers are self-inverse under A059897, the sequence thereby forms a subgroup of the positive integers under A059897. (End)
EXAMPLE
The Fermi-Dirac factorization of 160 is 2 * 5 * 16. The factors 2, 5 and 16 are A329050(0,0), A329050(2,0) and A329050(0,2), having symmetry about the main diagonal of A329050. So 160 is in the sequence.
PROG
(PARI) A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n)); ff(fa) = {for (i=1, #fa~, my(p=fa[i, 1]); fa[i, 1] = A019565(fa[i, 2]); fa[i, 2] = 2^(primepi(p)-1); ); fa; } \\ A225546pos(k, fs) = for (i=1, #fs, if (fs[i] == k, return(i)); );
normalize(f) = {my(list = List()); for (k=1, #f~, my(fk = factor(f[k, 1])); for (j=1, #fk~, listput(list, fk[j, 1])); ); my(fs = Set(list)); my(m = matrix(#fs, 2)); for (i=1, #m~, m[i, 1] = fs[i]; for (k=1, #f~, m[i, 2] += valuation(f[k, 1], fs[i])*f[k, 2]; ); ); m; }
isok(n) = my(fa=factor(n), fb=ff(fa)); normalize(fb) == fa; \\ Michel Marcus, Aug 05 2022
2, 3, 3, 5, 5, 5, 7, 7, 7, 7, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31
COMMENTS
Seen as a square array read by antidiagonals, a subtable of the binary operation multiplication tables A297845, A306697 and A329329. - Peter Munn, Jan 15 2020
REFERENCES
Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought", Basic Books, 1995.
FORMULA
From Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006: (Start)
a(n) = prime(floor(1/2 + sqrt(2*n))).
When viewed as a square array A(n,k), the following hold for n >= 1, k >= 1:
A(n,k) = prime(n+k-1).
A(n,1) = A(1,n) = prime(n), where prime(n) = A000040(n). A(n+1,k) = A(n,k+1) = A003961(A(n,k)). A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)) = A329329(A(n,1), A(1,k)). (End)
EXAMPLE
Triangle begins:
2;
3, 3;
5, 5, 5;
7, 7, 7, 7;
...
MATHEMATICA
Table[Prime[Floor[1/2 + Sqrt[2*n]]], {n, 1, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
Flatten[Table[Table[Prime[n], {n}], {n, 12}]] (* Alonso del Arte, Jan 18 2012 *) Table[PadRight[{}, n, Prime[n]], {n, 15}]//Flatten (* Harvey P. Dale, Feb 29 2024 *)
PROG
(Haskell)
a005145 n k = a005145_tabl !! (n-1) !! (k-1)
a005145_row n = a005145_tabl !! (n-1)
a005145_tabl = zipWith ($) (map replicate [1..]) a000040_list
a005145_list = concat a005145_tabl
(Magma) [NthPrime(Round(Sqrt(2*n))): n in [1..60]]; // Vincenzo Librandi, Jan 18 2020 (Python)
from sympy import primerange
a = []; [a.extend([pn]*n) for n, pn in enumerate(primerange(1, 32), 1)]
a(n) = prime(n)*prime(n+1)*...*prime(2*n-1), where prime(i) is the i-th prime.
+10
3
1, 2, 15, 385, 17017, 1062347, 86822723, 10131543907, 1204461778591, 198229051666003, 35224440615606707, 6295457783127226289, 1331590860773071702483, 310692537866322378582047, 78832548083496383033878901, 21381953681344611984282084241
COMMENTS
For n >= 0, a(n+1) is the n-th power of 15 in the monoid defined by A306697. - Peter Munn, Feb 18 2020
EXAMPLE
a(1)=2; a(2) = 3*5 = 15; a(3) = 5*7*11 = 385.
MAPLE
seq(mul(ithprime(n+k), k=0..n-1), n=0..15); # Muniru A Asiru, Mar 16 2019
MATHEMATICA
Table[Times@@Prime[Range[n, 2n-1]], {n, 20}] (* Harvey P. Dale, Jul 19 2018 *)
PROG
(Haskell)
(GAP) P:=Filtered([1..200], IsPrime);;
a:=List([1..15], n->Product([0..n-1], k->P[n+k])); # Muniru A Asiru, Mar 16 2019 (PARI) a(n) = prod(k=n, 2*n-1, prime(k)); \\ Michel Marcus, Mar 16 2019
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