Displaying 1-10 of 48 results found.
Primes congruent to 1 (mod 12).
+10
113
13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349, 373, 397, 409, 421, 433, 457, 541, 577, 601, 613, 661, 673, 709, 733, 757, 769, 829, 853, 877, 937, 997, 1009, 1021, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297
COMMENTS
This has several equivalent definitions (cf. the Tunnell link)
Also primes of the form x^2 + 9y^2 (discriminant -36). - T. D. Noe, May 07 2005 [corrected by Klaus Purath, Jan 18 2023] Also primes of the form x^2 + 4*x*y + y^2.
Also primes of the form x^2 + 2*x*y - 2*y^2 (cf. A084916). Also primes of the form x^2 + 6*x*y - 3*y^2.
Also primes of the form 4*x^2 + 8*x*y + y^2.
Also primes of the form u^2 - 3v^2 (use the transformation {u,v} = {x+2y,y}). - Tito Piezas III, Dec 28 2008 Sequence lists generalized cuban primes ( A007645) that are the sum of 2 nonzero squares. - Altug Alkan, Nov 25 2015 Yasutoshi Kohmoto observes that prevprime(a(n)) is more frequently congruent to 3 (mod 4) than to 1. This bias can be explained by the possible prime constellations and gaps: To have the same residue mod 4 as a prime in the list, the previous prime must be at a gap of 4 or 8 or 12 ..., but a gap of 4 is impossible because 12k + 1 - 4 is divisible by 3, and gaps >= 12 are very rare for small primes. To have the residue 3 (mod 4) the previous prime can be at a gap of 2 or 6 with no a priori divisibility property. However, this bias tends to disappear as the primes (and average prime gaps) grow bigger: for primes < 10^5, the ratio is about 35% vs. 65% as the above simple explanation suggests, but considering primes up to 10^8 yields a ratio of about 41% vs. 59%. It can be expected that the ratio asymptotically tends to 1:1. - M. F. Hasler, Sep 01 2017 Also primes of the form x^2 - 27*y^2. - Klaus Purath, Jan 18 2023
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
MAPLE
select(isprime, [seq(i, i=1..10000, 12)]); # Robert Israel, Nov 27 2015
MATHEMATICA
Select[Prime/@Range[250], Mod[ #, 12]==1&]
Select[Range[13, 10^4, 12], PrimeQ] (* Zak Seidov, Mar 21 2011 *)
PROG
(PARI) for(i=1, 250, if(prime(i)%12==1, print(prime(i))))
(PARI) forstep(p=13, 10^4, 12, isprime(p)&print(p)); \\ Zak Seidov, Mar 21 2011 (Magma) [p: p in PrimesUpTo(1400) | p mod 12 in {1}]; // Vincenzo Librandi, Jul 14 2012 For other programs see the "Binary Quadratic Forms and OEIS" link.
CROSSREFS
Cf. A068227, A068229, A040117, A068231, A068232, A068233, A068234, A068235, A139643, A141122, A140633, A264732. For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
AUTHOR
Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002
EXTENSIONS
Entry revised by N. J. A. Sloane, Oct 18 2014 (Edited, merged with A141122, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008).
Primes of form 8n+1, that is, primes congruent to 1 mod 8.
(Formerly M5037)
+10
100
17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 617, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1033, 1049, 1097, 1129, 1153, 1193, 1201, 1217, 1249, 1289, 1297, 1321, 1361
COMMENTS
Discriminant is 32, class is 2. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Integers n (n > 9) of form 4k + 1 such that binomial(n-1, (n-1)/4) == 1 (mod n) - Benoit Cloitre, Feb 07 2004 Primes of the form x^2 + 8y^2. - T. D. Noe, May 07 2005 Is this the same sequence as A141174? Being a subset of A001132 and also a subset of A038873, this is also a subset of the primes of the form u^2 - 2v^2. - Tito Piezas III, Dec 28 2008 These primes p are only which possess the property: for every integer m from interval [0, p) with the Hamming distance D(m, p) = 2, there exists an integer h from (m, p) with D(m, h) = 2. - Vladimir Shevelev, Apr 18 2012 Primes p such that p XOR 6 = p + 6. - Brad Clardy, Jul 22 2012 Odd primes p such that -1 is a 4th power mod p. - Eric M. Schmidt, Mar 27 2014 There are infinitely many primes of this form. See Brubaker link. - Alonso del Arte, Jan 12 2017 These primes split in Z[sqrt(2)]. For example, 17 = (-1)(1 - 3*sqrt(2))(1 + 3*sqrt(2)). This is also true of primes of the form 8n - 1. - Alonso del Arte, Jan 26 2017
REFERENCES
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
Z. I. Borevich and I. R. Shafarevich, Number Theory.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
EXAMPLE
a(1) = 17 = 2 * 8 + 1 = (10001)_2. All numbers m from [0, 17) with the Hamming distance D(m, 17) = 2 are 0, 3, 5, 9. For m = 0, we can take h = 3, since 3 is drawn from (0, 17) and D(0, 3) = 2; for m = 3, we can take h = 5, since 5 from (3, 17) and D(3, 5) = 2; for m = 5, we can take h = 6, since 6 from (5, 17) and D(5, 6) = 2; for m = 9, we can take h = 10, since 10 is drawn from (9, 17) and D(9, 10) = 2. - Vladimir Shevelev, Apr 18 2012
PROG
(Haskell)
a007519 n = a007519_list !! (n-1)
a007519_list = filter ((== 1) . a010051) [1, 9..]
(Magma) [p: p in PrimesUpTo(2000) | p mod 8 eq 1 ]; // Vincenzo Librandi, Aug 21 2012 (PARI) lista(nn)= my(vpr = []); for (x = 0, nn, y = 0; while ((v = x^2+6*x*y+y^2) < nn, if (isprime(v), if (! vecsearch(vpr, v), vpr = concat(vpr, v); vpr = vecsort(vpr); ); ); y++; ); ); vpr; \\ Michel Marcus, Feb 01 2014 (SageMath) # uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 4, -4])
print(Q.represented_positives(1361, 'prime')) # Peter Luschny, Jan 26 2017
CROSSREFS
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1.
+10
96
1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, 89, 103, 118, 134, 151, 169, 188, 208, 229, 251, 274, 298, 323, 349, 376, 404, 433, 463, 494, 526, 559, 593, 628, 664, 701, 739, 778, 818, 859, 901, 944, 988, 1033, 1079, 1126, 1174, 1223, 1273, 1324, 1376, 1429, 1483
COMMENTS
Number of 1's in the n X n lower Hessenberg (0,1)-matrix (i.e., the matrix having 1's on or below the superdiagonal and 0's above the superdiagonal).
If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-2) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007 Number of binary operations which have to be added to Moisil's algebras to obtain algebraic counterparts of n-valued Łukasiewicz logics. See the Wójcicki and Malinowski book, page 31. - Artur Jasinski, Feb 25 2010 Also (n + 1)!(-1)^(n + 1) times the determinant of the n X n matrix given by m(i,j) = i/(i+1) if i=j and otherwise 1. For example, (5+1)! * ((-1)^(5+1)) * Det[{{1/2, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1}, {1, 1, 3/4, 1, 1}, {1, 1, 1, 4/5, 1}, {1, 1, 1, 1, 5/6}}] = 19 = a(5), and (6+1)! * ((-1)^(6+1)) * Det[{{1/2, 1, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1, 1}, {1, 1, 3/4, 1, 1, 1}, {1, 1, 1, 4/5, 1, 1}, {1, 1, 1, 1, 5/6, 1}, {1, 1, 1, 1, 1, 6/7}}] = 26 = a(6). - John M. Campbell, May 20 2011 2*a(n-1) = n*(n+1) - 4, n>=0, with a(-1) = -2 and a(0) = -1, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 17 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013 With a(0) = 1 and a(1) = 2, a(n-1) is the number of distinct values of 1 +- 2 +- 3 +- ... +- n, for n > 0. - Derek Orr, Mar 11 2015 Also, numbers m such that 8*m+17 is a square. - Bruno Berselli, Sep 16 2015 Omar E. Pol's formula from Apr 23 2008 can be interpreted as the position of an element located on the third diagonal of an triangular array (read by rows) provided n > 1. - Enrique Pérez Herrero, Aug 29 2016
a(n) is the sum of the numerator and denominator of the fraction that is the sum of 2/(n-1) + 2/n; all fractions are reduced and n > 2. - J. M. Bergot, Jun 14 2017 a(n) is also the number of maximal irredundant sets in the (n+2)-path complement graph for n > 1. - Eric W. Weisstein, Apr 12 2018 a(n) is not divisible by primes listed in A038890. The prime factors are given in A038889 and the prime terms of the sequence are listed in A124199. Each odd prime factor p divides exactly 2 out of any p consecutive terms with the exception of 17, which appears only once in such an interval of terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -3 (mod p), see examples.
If A is a sequence satisfying the recurrence t(n) = 5*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 4 or A(0) = -1, A(1) = n-1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i for i>0. (End)
Mark each point on a 4^n grid with the number of points that are visible from the point; for n > 1, a(n) is the number of distinct values in the grid. - Torlach Rush, Mar 23 2021
REFERENCES
A. S. Karpenko, Łukasiewicz's Logics and Prime Numbers, 2006 (English translation).
G. C. Moisil, Essais sur les logiques non-chrysippiennes, Ed. Academiei, Bucharest, 1972.
Wójcicki and Malinowski, eds., Łukasiewicz Sentential Calculi, Wrocław: Ossolineum, 1977.
LINKS
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, Vol. 35, No. 4 (1997), pp. 318-328.
FORMULA
G.f.: A(x) = x*(1 + x - x^2)/(1 - x)^3.
a(n) = binomial(n+2, 2) - 2. - Paul Barry, Feb 27 2003 With offset 5, this is binomial(n, 0) - 2*binomial(n, 1) + binomial(n, 2), the binomial transform of (1, -2, 1, 0, 0, 0, ...). - Paul Barry, Jul 01 2003 Binomial transform of (1, 3, 1, 0, 0, 0, ...). Also equals A130296 * [1,2,3,...]. - Gary W. Adamson, Jul 27 2007 a(n) = a(n-1) + n + 1 for n >= 1.
a(n) = n*(n-1)/2 + 2*n - 1.
a(n) = Hyper2F1([-2, n-1], [1], -1). - Peter Luschny, Aug 02 2014 For n > 1, a(n) = 4*binomial(n-1,1) + binomial(n-2,2), comprising the third column of A267633. - Tom Copeland, Jan 25 2016 Sum_{n>=1} 1/a(n) = 3/2 + 2*Pi*tan(sqrt(17)*Pi/2)/sqrt(17). - Amiram Eldar, Jan 06 2021
EXAMPLE
By the definition (first formula):
----------------------------------------------------------------------
1 4 8 13 19 26
----------------------------------------------------------------------
X
X X X
X X X X X X
X X X X X X X X X X
X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X
----------------------------------------------------------------------
(End)
Assuming a(i) is divisible by p with 0 < i < p and a(k) is the next term divisible by p, then from i + k == -3 (mod p) follows that k = min(p*m - i - 3) != i for any integer m.
(1) 17|a(7) => k = min(17*m - 10) != 7 => m = 2, k = 24 == 7 (mod 17). Thus every a(17*m + 7) is divisible by 17.
(2) a(9) = 53 => k = min(53*m - 12) != 9 => m = 1, k = 41. Thus every a(53*m + 9) and a(53*m + 41) is divisible by 53.
(3) 101|a(273) => 229 == 71 (mod 101) => k = min(101*m - 74) != 71 => m = 1, k = 27. Thus every a(101*m + 27) and a(101*m + 71) is divisible by 101. (End)
Illustration of initial terms: _ _
. _ _ |_|_|_
. _ _ |_|_|_ |_|_|_|_
. _ _ |_|_|_ |_|_|_|_ |_|_|_|_|_
. _ _ |_|_|_ |_|_|_|_ |_|_|_|_|_ |_|_|_|_|_|_
. _ |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
. |_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
.
. 1 4 8 13 19 26
------------------------------------------------------------------------ (End)
MAPLE
a := n -> hypergeom([-2, n-1], [1], -1);
MATHEMATICA
f[n_] := n (n + 3)/2 - 1; Array[f, 55] (* or *) k = 2; NestList[(k++; # + k) &, 1, 55] (* Robert G. Wilson v, Jun 11 2010 *) Table[Binomial[n + 1, 2] + n - 1, {n, 53}] (* or *)
Rest@ CoefficientList[Series[x (1 + x - x^2)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Aug 29 2016 *)
PROG
(Magma) [Binomial(n + 1, 2) + n - 1: n in [1..60]]; // Vincenzo Librandi, May 21 2011 n-1+(n+1)*n/2
(Haskell)
CROSSREFS
Triangular numbers ( A000217) minus two. a(n) = T(3, n-2), array T as in A049600. Third diagonal of triangle in A059317.
Primes of the form 8n+7, that is, primes congruent to -1 mod 8.
(Formerly M4376)
+10
82
7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263, 271, 311, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 887, 911, 919, 967, 983, 991, 1031, 1039, 1063, 1087, 1103, 1151
COMMENTS
Primes that are the sum of no fewer than four positive squares.
Discriminant is 32, class is 2. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Primes p such that x^4 = 2 has just two solutions mod p. Subsequence of A040098. Solutions mod p are represented by integers from 0 to p - 1. For p > 2, i is a solution mod p of x^4 = 2 if and only if p - i is a solution mod p of x^4 = 2, so the sum of the two solutions is p. The solutions are given in A065907 and A065908. - Klaus Brockhaus, Nov 28 2001 As this is a subset of A001132, this is also a subset of the primes of form x^2 - 2y^2. And as this is also a subset of A038873, this is also a subset of the primes of form x^2 - 2y^2. - Tito Piezas III, Dec 28 2008 Primes p such that p XOR 6 = p - 6. - Brad Clardy, Jul 22 2012
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
LINKS
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
MAPLE
select(isprime, [seq(i, i=7..10000, 8)]); # Robert Israel, Nov 22 2016
PROG
(PARI) A007522(m) = local(p, s, x, z); forprime(p = 3, m, s = []; for(x = 0, p-1, if(x^4%p == 2%p, s = concat(s, [x]))); z = matsize(s)[2]; if(z == 2, print1(p, ", "))) A007522(1400) (Haskell)
a007522 n = a007522_list !! (n-1)
a007522_list = filter ((== 1) . a010051) a004771_list
Primes congruent to {0, 1, 4} mod 5.
+10
76
5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509, 521, 541, 569, 571, 599, 601, 619
COMMENTS
Also odd primes p such that 5 is a square mod p: (5/p) = +1 for p > 5.
Primes of the form x^2 + x*y - y^2 (as well as of the form x^2 + 3*x*y + y^2), both with discriminant = 5 and class number = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1. [This was originally a separate entry, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales, Jun 06 2008. R. J. Mathar proved on Jul 22 2008 that this coincides with the present sequence.] Also primes of the form x^2 + 4*x*y - y^2. Every binary quadratic primitive form of discriminant 20 or 5 has proper solutions for positive integers N given in A089270, including the present primes. Proof from computing the corresponding representative parallel primitive forms, which leads to x^2 - 5 == 0 (mod N) or x^2 + x - 1 == 0 (mod N) which have solutions precisely for these positive N values, including these primes. - Wolfdieter Lang, Jun 19 2019 For a Pythagorean triple a, b, c, these primes (and 2) are the possible prime factors of 2a + b, |2a - b|, 2b + a, and 2b - a. - J. Lowell, Nov 05 2011 Except for p = 5, these are primes p that divide Fibonacci(p-1). - Dmitry Kamenetsky, Jul 27 2015 Apart from the first term, these are rational primes that decompose in the field Q[sqrt(5)]. For example, 11 = ((7 + sqrt(5))/2)*((7 - sqrt(5))/2), 19 = ((9 + sqrt(5))/2)*((9 - sqrt(5))/2). - Jianing Song, Nov 23 2018
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
MAPLE
select(isprime, [5, seq(op([5*k-1, 5*k+1]), k=1..1000)]); # Robert Israel, Dec 22 2014
MATHEMATICA
Join[{5}, Select[Prime[Range[4, 100]], Mod[#, 5] == 1 || Mod[#, 5] == 4 &]] (* Alonso del Arte, Nov 27 2011 *)
PROG
(Magma) [ p: p in PrimesUpTo(700) | p mod 5 in {0, 1, 4}]; // Vincenzo Librandi, Aug 21 2012 (GAP) Filtered(Concatenation([5], Flat(List([1..140], k->[5*k-1, 5*k+1]))), IsPrime); # Muniru A Asiru, Nov 24 2018
EXTENSIONS
Corrected and extended by Peter K. Pearson, May 29 2005
Odd primes p such that 13 is a square mod p.
+10
47
3, 13, 17, 23, 29, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139, 157, 173, 179, 181, 191, 199, 211, 233, 251, 257, 263, 269, 277, 283, 311, 313, 337, 347, 367, 373, 389, 419, 433, 439, 443, 467, 491, 503, 521, 523, 547, 563, 569, 571, 599, 601, 607, 641
COMMENTS
Equivalently, by quadratic reciprocity (since 13 == 1 (mod 4)), primes p which are squares mod 13.
The squares mod 13 are 0, 1, 4, 9, 3, 12 and 10.
Also primes of the form x^2 + 3*x*y - y^2. Discriminant = 13. Class = 1. This was originally a separate entry, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 06 2008. R. J. Mathar proved that this coincides with the present sequence, Jul 22 2008 Primes p such that x^2 + x = 3 has a solution mod p (the solutions over the reals are (-1+-sqrt(13))/2). - Joerg Arndt, Jul 27 2011
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
EXAMPLE
13 == 1 (mod 3) and 1 is a square, so 3 is on the list.
101 is prime and congruent to 7^2 = 49 == 10 (mod 13), so 101 is on the list.
MATHEMATICA
Select[ Prime@ Range@ 118, JacobiSymbol[ #, 13] > -1 &] (* Robert G. Wilson v, May 16 2008 *) Select[Flatten[Table[13n + {1, 3, 4, 9, 10, 12}, {n, 50}]], PrimeQ[#] &] (* Alonso del Arte, Sep 16 2012 *)
PROG
(PARI) select( {is_ A038883(n)=bittest(5659, n%13)&&isprime(n)}, [0..666]) \\ M. F. Hasler, Feb 17 2022 (Sage) # uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 3, -1])
print(Q.represented_positives(641, 'prime')) # Peter Luschny, Sep 20 2018
CROSSREFS
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Primes congruent to 1 or 19 (mod 30).
+10
42
19, 31, 61, 79, 109, 139, 151, 181, 199, 211, 229, 241, 271, 331, 349, 379, 409, 421, 439, 499, 541, 571, 601, 619, 631, 661, 691, 709, 739, 751, 769, 811, 829, 859, 919, 991, 1009, 1021, 1039, 1051, 1069, 1129, 1171, 1201, 1231, 1249, 1279, 1291, 1321, 1381
COMMENTS
Theorem: Same as primes of the form x^2+15*y^2 (discriminant -60). Proof: Cox, Cor. 2.27, p. 36.
Equivalently, primes congruent to 1 or 4 (mod 15). Also x^2+xy+4y^2 is the principal form of (fundamental) discriminant -15. The only other class for -15 contains the form 2x^2+xy+2y^2 ( A106859), in the other genus. - Rick L. Shepherd, Jul 25 2014 Three further theorems (these were originally stated as conjectures, but are now known to be theorems, thanks to the work of J. B. Tunnell - see link):
1. The same as primes of the form x^2-xy+4y^2 (discriminant -15) and x^2-xy+19y^2 (discriminant -75), both with x and y nonnegative. - T. D. Noe, Apr 29 2008 2. The same as primes of the form x^2+xy+19y^2 (discriminant -75), with x and y nonnegative. - T. D. Noe, Apr 29 2008 3. The same as primes of the form x^2+5xy-5y^2 (discriminant 45). - N. J. A. Sloane, Jun 01 2014 Also primes of the form x^2+7*x*y+y^2 (discriminant 45).
Lemma ( Will Jagy, Jun 12 2014): If c is any (positive or negative) even number, then x^2 + x y + c y^2 and x^2 + (4 c - 1) y^2 represent the same odd numbers. Proof: x (x + y) + c y^2 = odd, therefore x is odd, x + y odd, so y is even. Let y = 2 t. Then x( x + 2 t) + 4 c t^2 = x^2 + 2 x t + 4 c t^2 = (x+t)^2 + (4c-1) t^2 = odd. QED With c = 4, neither one represents 2, so x^2+15y^2 and x^2+xy+4y^2 represent the same primes.
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
MATHEMATICA
QuadPrimes2[1, 0, 15, 10000] (* see A106856 *) Select[Prime@Range[250], MemberQ[{1, 19}, Mod[#, 30]] &] (* Vincenzo Librandi, Apr 05 2015 *)
PROG
(PARI) is(p)=issquare(Mod(p, 15))&&isprime(p) \\ M. F. Hasler, Jan 15 2016
CROSSREFS
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
EXTENSIONS
Edited by N. J. A. Sloane, Jun 01 2014 and Oct 18 2014: added Tunnell document, revised entry, merged with A141184. The latter entry was submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008.
Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 17.
+10
24
1, 2, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 2, 0, 0, 5, 1, 2, 2, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 6, 0, 2, 0, 3, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 2, 0, 6, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 7, 0, 0, 2, 3, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 1
COMMENTS
Coefficients of Dedekind zeta function for the quadratic number field of discriminant 17. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022
FORMULA
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(4+sqrt(17))/sqrt(17) = 1.016084... . - Amiram Eldar, Oct 11 2022 a(n) = Sum_{d|n} Kronecker(17, d).
Multiplicative with a(17^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(17, p) = -1 (p is in A038890), and a(p^e) = e+1 if Kronecker(17, p) = 1 (p is in A038889 \ {17}). (End)
MATHEMATICA
a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[17, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
PROG
(PARI) my(m=17); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(17, d)); \\ Amiram Eldar, Nov 18 2023
CROSSREFS
Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively. Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Absolute value of numerator of determinant of n X n matrix with M(i,j) = i/(i+1) if i=j otherwise 1.
+10
17
1, 2, 1, 13, 19, 13, 17, 43, 53, 1, 19, 89, 103, 59, 67, 151, 13, 47, 1, 229, 251, 137, 149, 1, 349, 47, 101, 433, 463, 1, 263, 43, 593, 157, 83, 701, 739, 389, 409, 859, 53, 59, 1, 1033, 83, 563, 587, 1223, 67, 331, 1, 1429, 1483, 769, 797, 127, 1709, 1, 457, 1889
COMMENTS
Numbers n such that a(n) = 1 are listed in A127852. All a(n)>1 are prime belonging to A038889 (i.e., 17 is a square mod a(n)).
FORMULA
det(M) = (-1)^(n+1)*(n^2+3*n-2)/(2*(n+1)!), implying that a(n)=p, where p= A006530(n^2+3*n-2) is the largest prime divisor of (n^2+3*n-2), if p>n+1 or p=sqrt((n^2+3*n-2)/2); otherwise a(n)=1. a(n) = Numerator[(-1)^(n+1) Det[ DiagonalMatrix[ Table[ i/(i+1) - 1, {i, 1, n} ] ] + 1 ]].
a(n) = Numerator[ (n^2+3n-2)/(2(n+1)!) ] = Numerator[ ((2n+3)^2-17)/(4(n+1)!) ].
MATHEMATICA
Numerator[Table[(-1)^(n+1) Det[ DiagonalMatrix[ Table[ i/(i+1) - 1, {i, 1, n} ] ] + 1 ], {n, 1, 70} ]]
Table[ Numerator[ (n^2+3n-2)/(2(n+1)!) ], {n, 1, 100} ]
Primes p which have Kronecker symbol (p|34) = 1.
+10
16
3, 5, 11, 29, 37, 47, 61, 89, 103, 107, 109, 127, 131, 137, 139, 151, 163, 173, 181, 191, 197, 211, 223, 227, 239, 257, 263, 269, 271, 277, 281, 283, 317, 347, 353, 359, 379, 383, 397, 409, 419, 433, 457, 463, 499, 541, 547, 569, 571, 577, 593, 599, 619, 631
COMMENTS
Originally incorrectly named "primes which are squares (mod 34)", which is sequence A038889. - M. F. Hasler, Jan 15 2016
MATHEMATICA
Select[Prime[Range[200]], JacobiSymbol[#, 34]==1&]
PROG
(Magma) [p: p in PrimesUpTo(631) | KroneckerSymbol(p, 34) eq 1]; // Vincenzo Librandi, Sep 11 2012
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