| Displaying 1-7 of 7 results found.
page
1
Prime numbers that are the sum of the divisors of some n.
+10
29
3, 7, 13, 31, 127, 307, 1093, 1723, 2801, 3541, 5113, 8011, 8191, 10303, 17293, 19531, 28057, 30103, 30941, 86143, 88741, 131071, 147073, 292561, 459007, 492103, 524287, 552793, 579883, 598303, 684757, 704761, 732541, 735307, 797161, 830833, 1191373
COMMENTS
If n > 2 and sigma(n) is prime, then n must be an even power of a prime number. For example, 1093 = sigma(3^6). - T. D. Noe, Jan 20 2004 All primes of the form 2^n-1 (Mersenne primes) are in the sequence because if n is a natural number then sigma(2^(n-1)) = 2^n-1. So A000668 is a subsequence of this sequence. If sigma(n) is prime then n is of the form p^(q-1) where both p & q are prime (the proof is easy). - Farideh Firoozbakht, May 28 2005 Primes of the form 1 + p + p^2 + ... + p^k where p is prime.
Except for 3, these primes are particular Brazilian primes belonging to A085104. These prime numbers are also Brazilian primes of the form (p^x - 1)/(p^y - 1), p prime, belonging to A003424, with here x is prime, and y = 1. [See section V.4 of Quadrature article in Links.] - Bernard Schott, Dec 25 2012 Others subsequences of this sequence:
A053183 for 111_p = p^2 + p + 1 when p is prime.
A190527 for 11111_p = p^4 + p^3 + p^2 + p + 1 when p is prime.
A194257 for 1111111_p = p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime. (End)
LINKS
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
EXAMPLE
307 = 1 + 17 + 17^2; 307 and 17 are primes.
MATHEMATICA
t={3}; lim=10^9; n=1; While[p=Prime[n]; k=2; s=1+p+p^2; s<lim, While[s<lim, If[PrimeQ[s], AppendTo[t, s]]; k=k+2; s=s+(1+p)p^(k-1)]; n++]; t=Union[t]
Select[DivisorSigma[1, Range[2 10^6]], PrimeQ]//Union (* Harvey P. Dale, Jun 18 2022 *)
PROG
(PARI) upto(lim)=my(v=List([3]), t); forprime(p=2, solve(x=1, lim^(1/4), x^4+x^3+x^2+x+1-lim), forprime(e=5, 1+log(lim)\log(p), if(isprime(t=sigma(p^(e-1))) && t<=lim, listput(v, t)))); forprime(p=2, solve(x=1, lim^(1/2), x^2+x+1-lim), if(isprime(t=p^2+p+1), listput(v, t))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Dec 20 2011 (Python)
from sympy import isprime, divisor_sigma
A023195_list = sorted(set([3]+[n for n in (divisor_sigma(d**2) for d in range(1, 10**4)) if isprime(n)])) # Chai Wah Wu, Jul 23 2016
a(n) = 1 + prime(n) + prime(n)^2 + prime(n)^3 + prime(n)^4.
+10
11
31, 121, 781, 2801, 16105, 30941, 88741, 137561, 292561, 732541, 954305, 1926221, 2896405, 3500201, 4985761, 8042221, 12326281, 14076605, 20456441, 25774705, 28792661, 39449441, 48037081, 63455221, 89451461, 105101005, 113654321
COMMENTS
Giovanni Resta has found that 28792661 is the first Sophie Germain prime of this form (and actually of the form p = (n^m-1)/(n-1) for any p-1 > n, m > 1). - M. F. Hasler, Mar 03 2020
REFERENCES
Victor Thébault, Curiosités arithmétiques, Mathesis 62 (1953), pp. 120-129.
FORMULA
a(n) = (prime(n)^5 - 1)/(prime(n) - 1) = A053699(prime(n)). (This is also meant by the 2nd formula.) - M. F. Hasler, Mar 03 2020
EXAMPLE
a(1) = 31 because prime(1) = 2 and 1 + 2 + 2^2 + 2^3 + 2^4 = 1 + 2 + 4 + 8 + 16 = 31.
MATHEMATICA
Table[Sum[Prime[n]^k, {k, 0, 4}], {n, 30}] (* Alonso del Arte, May 24 2015 *)
CROSSREFS
Equals A053699 restricted to prime indices. Subsequence of primes is A190527.
Primes p such that p^4 + p^3 + p^2 + p + 1 is prime.
+10
7
2, 7, 13, 17, 23, 29, 43, 73, 79, 83, 127, 193, 227, 239, 263, 277, 337, 359, 373, 397, 439, 457, 479, 503, 557, 563, 617, 919, 967, 1009, 1129, 1187, 1249, 1297, 1327, 1429, 1493, 1553, 1579, 1657, 1663, 1979, 1987, 2069, 2243, 2383, 2617, 2663, 2789
EXAMPLE
a(4) = 17 because 17 is prime and 17^4 + 17^3 + 17^2 + 17 + 1 = 88741 is prime.
MATHEMATICA
Select[Prime[Range[500]], PrimeQ[Total[#^Range[0, 4]]]&] (* Harvey P. Dale, Apr 08 2017 *)
PROG
(PARI) { n=0; for (m=1, 10^9, p=prime(m); if (isprime(p^4 + p^3 + p^2 + p + 1), write("b065509.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 20 2009 (PARI) { A065509_vec(N, p=1)=vector(N, i, until(isprime((p^5-1)\(p-1)), p=nextprime(p+1)); p)} \\ M. F. Hasler, Mar 03 2020 (Magma) [n: n in [0..10000]| IsPrime(n) and IsPrime(n^4+n^3+n^2+n+1)] // Vincenzo Librandi, Aug 07 2010
Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1 where n is not prime.
+10
7
43, 73, 157, 211, 241, 421, 463, 601, 757, 1123, 1483, 2551, 2971, 3307, 3907, 4423, 4831, 5701, 6007, 6163, 6481, 8191, 9901, 11131, 12211, 12433, 13807, 14281, 19183, 20023, 20593, 21757, 22621, 22651, 23563, 24181, 26083, 26407, 27061, 28393, 31153, 35533
COMMENTS
These numbers are Brazilian primes belonging to A085104. Brazilian primes with n prime are A023195, except 3 which is not Brazilian. k + 1 is necessarily prime, but that's not sufficient: 1 + 10 + 100 = 111.
Most of these terms come from A185632 which are prime numbers 111_n with n no prime, the first other term is 22621 = 11111_12, the next one is 245411 = 11111_22. Number of terms < 10^k: 0, 2, 9, 23, 64, 171, 477, 1310, 3573, 10098, ..., . - Robert G. Wilson v, Apr 15 2017
LINKS
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, April-June 2010, pages 30-38, included here with permission from the editors of Quadrature.
EXAMPLE
157 = 12^2 + 12 + 1 = 111_12 is prime and 12 is composite.
MAPLE
N:= 40000: # to get all terms <= N
res:= NULL:
for k from 2 to ilog2(N) do if isprime(k) then
for n from 2 do
p:= (n^(k+1)-1)/(n-1);
if p > N then break fi;
if isprime(p) and not isprime(n) then res:= res, p fi
od fi od:
res:= {res}:
MATHEMATICA
mx = 36000; g[n_] := Select[Drop[Accumulate@Table[n^ex, {ex, 0, Log[n, mx]}], 2], PrimeQ]; k = 1; lst = {}; While[k < Sqrt@mx, If[CompositeQ@k, AppendTo[lst, g@k]; lst = Sort@Flatten@lst]; k++]; lst (* Robert G. Wilson v, Apr 15 2017 *)
PROG
(PARI) isok(n) = {if (isprime(n), forcomposite(b=2, n, d = digits(n, b); if ((#d > 2) && (vecmin(d) == 1) && (vecmax(d)== 1), return(1)); ); ); return(0); } \\ Michel Marcus, Apr 09 2017 (PARI) A285017_vec(n)={my(h=vector(n, i, 1), y, c, z=4, L:list); L=List(); forprime(x=3, , forcomposite(m=z, x-1, y=digits(x, m); if((y==h[1..#y])&&2<#y, listput(L, x); z=m; if(c++==n, return(Vec(L))))))} \\ R. J. Cano, Apr 18 2017
Smallest divisor of sigma(n) that does not divide n.
+10
5
3, 2, 7, 2, 4, 2, 3, 13, 3, 2, 7, 2, 3, 2, 31, 2, 13, 2, 3, 2, 3, 2, 5, 31, 3, 2, 8, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 31, 3, 3, 2, 7, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 127, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 4, 2, 3
COMMENTS
a(n) = 2 iff n is an odd number that is not a perfect square.
(1) Every a(n) > n is a prime: Because of the minimality of a(n), a(n) = u*v with gcd(u,v)=1 leads to the contradiction (u*v)|n. Similarly, a(n)=p^k with p prime an k>1 leads to the contradiction (p^k-1)/(p-1) | n.
(2) n=p^(2*k), k>=1 and 2*k+1 prime, when a(n) = sigma(n) for n>2: Because n having two distinct prime factors implies sigma(n) composite, and if n is an odd power of a prime then 2|sigma(n). Finally, if 2*k+1=u*v with u,v > 1 then sigma(p^(u-1)) divides sigma(p^(2*k)), but not p^(2k), for any prime p, contradicting minimality of a(n). For example, no number sigma(p^8) for any prime p is in the sequence.
(3) The converse of (2) is false since, e.g. sigma(7^2) = 3*19 so that a(7^2) = 3, and sigma(2^10) = 23*89 so that a(2^10) = 23.
(4) Conjecture: a(n) > n implies a(n) = sigma(n); tested through n = 20000000.
(5) Subsequences are: A053183 (sigma(p^2) is prime for prime p), A190527 (sigma(p^4) is prime for prime p), A194257 (sigma(p^6) is prime for prime p), A286301 (sigma(p^10) is prime for prime p) (6) Subsequences are: A000668 (primes of form 2^p-1), A076481 (primes of form (3^p-1)/2), A086122 (primes of form (5^p-1)/4), A102170 (primes of form (7^p-1)/6), all when p is prime. (End)
Up to n = 10^6, there are 89 distinct elements. For those n, a(n) is prime. If it's not, it's a power of 2, a power of 3 or a perfect square <= 121. - David A. Corneth, May 10 2017
MATHEMATICA
a193574[n_] := First[Select[Divisors[DivisorSigma[1, n]], Mod[n, #]!=0&]]
PROG
(PARI) a(n)=local(ds); ds=divisors(sigma(n)); for(k=2, #ds, if(n%ds[k], return(ds[k])))
(Haskell)
import Data.List ((\\))
a193574 n = head [d | d <- [1..sigma] \\ nDivisors, mod sigma d == 0]
where nDivisors = a027750_row n
sigma = sum nDivisors
Primes of the form n^4 + n^3 + n^2 + n + 1 where n is nonprime.
+10
3
5, 22621, 245411, 346201, 637421, 837931, 2625641, 3835261, 6377551, 15018571, 16007041, 21700501, 30397351, 35615581, 52822061, 78914411, 97039801, 147753211, 189004141, 195534851, 209102521, 223364311, 279086341, 324842131, 421106401, 445120421, 566124791, 693025471, 727832821, 745720141, 880331261, 943280801, 987082981, 1544755411, 1740422941
COMMENTS
Note that there are no primes of the form n^3 + n^2 + n + 1 = (n+1)*(n^2+1) as irreducible components over Z.
These are the primes associated with A286094. All the numbers of this sequence n^4 + n^3 + n^2 + n + 1 = 11111_n with n > 1 are Brazilian numbers, so belong to A125134 and A085104. (End)
FORMULA
{n^4 + n^3 + n^2 + n + 1 where n is in A018252}.
EXAMPLE
a(1) = 1^4 + 1^3 + 1^2 + 1 + 1 = 5.
a(2) = 12^4 + 12^3 + 12^2 + 12 + 1 = 22621.
MAPLE
for n from 1 to 150 do p(n):= 1+n+n^2+n^3+n^4;
if tau(n)>2 and isprime(p(n)) then print(n, p(n)) else fi od: # Bernard Schott, May 15 2017
MATHEMATICA
Select[Map[Total[#^Range[0, 4]] &, Select[Range@ 204, ! PrimeQ@ # &]], PrimeQ] (* Michael De Vlieger, May 15 2017 *)
PROG
(PARI) print1(5); forcomposite(n=4, 1e3, if(isprime(t=n^4+n^3+n^2+n+1), print1(", "t))) \\ Charles R Greathouse IV, Mar 25 2013
Nonprime numbers n such that n^4 + n^3 + n^2 + n + 1 is prime.
+10
3
1, 12, 22, 24, 28, 30, 40, 44, 50, 62, 63, 68, 74, 77, 85, 94, 99, 110, 117, 118, 120, 122, 129, 134, 143, 145, 154, 162, 164, 165, 172, 175
COMMENTS
The corresponding prime numbers n^4 + n^3 + n^2 + n + 1 = 11111_n are in A193366; these Brazilian primes, except 5 which is not Brazilian, belong to A085104 and A285017.
EXAMPLE
12 is in the sequence because 12^4 + 12^3 + 12^2 + 12 + 1 = 11111_12 = 22621, which is prime.
MATHEMATICA
Select[Range@ 414, And[! PrimeQ@ #, PrimeQ[Total[#^Range[0, 4]]]] &] (* Michael De Vlieger, May 03 2017 *)
PROG
(PARI) isok(n)=if(n==1, 5, if(ispseudoprime(n), 0, isprime(fromdigits([1, 1, 1, 1, 1], n))));
getfirstterms(n)={my(L:list, c:small); L=List(); c=0; forstep(k=1, +oo, 1, if(isok(k), listput(L, k); if(c++==n, break))); return(Vec(L))} \\ R. J. Cano, May 09 2017
Search completed in 0.010 seconds
|