A003630 -id:A003630 - cp's OEIS frontend

A035194 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 12.

1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 2, 0, 0, 1, 2, 0, 2, 0, 0, 0, 0, 2, 0, 2, 2, 1, 1, 1, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 1, 2, 2, 1, 0, 0, 2, 0, 0, 1

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 12. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

G. C. Greubel, Table of n, a(n) for n = 1..10000

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.
Cf. A003630, A097933, A196530.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[12, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

my(m=12); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(12, d)); \\ Amiram Eldar, Nov 18 2023

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2+sqrt(3))/sqrt(3) = 0.760345... (A196530). - Amiram Eldar, Oct 11 2022
From Amiram Eldar, Nov 18 2023: (Start)
a(n) = Sum_{d|n} Kronecker(12, d).
Multiplicative with a(p^e) = 1 if Kronecker(12, p) = 0 (p = 2 or 3), a(p^e) = (1+(-1)^e)/2 if Kronecker(12, p) = -1 (p is in A003630), and a(p^e) = e+1 if Kronecker(12, p) = 1 (p is in A097933). (End)

A097933 Primes p that divide 3^((p-1)/2) - 1.

Original entry on oeis.org

11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601, 613

Offset: 1

Cino Hilliard, Sep 04 2004

nonn

Rational primes that decompose in the field Q[sqrt(3)]. - N. J. A. Sloane, Dec 26 2017
For all primes p > 2 and integers gcd(x, y, p) = 1, x^((p-1)/2) +- y^((p-1)/2) is divisible by p. This is because (x^((p-1)/2) - y^((p-1)/2))(x^((p-1)/2) + y^((p-1)/2)) = x^(p-1) - y^(p-1) is divisible by p according to Fermat's Little Theorem (FLT). This sequence lists p that divides 3^((p-1)/2) - 1^((p-1)/2), and A003630 lists the '+' case.
Apart from initial terms, this and A038874 are the same. - N. J. A. Sloane, May 31 2009
Primes in A091998. - Reinhard Zumkeller, Jan 07 2012
Also, primes congruent to 1 or 11 (mod 12). - Vincenzo Librandi, Mar 23 2013
Conjecture: Let r(n) = (a(n) - 1)/(a(n) + 1) if a(n) mod 4 = 1, (a(n) + 1)/(a(n) - 1) otherwise; then Product_{n>=1} r(n) = (6/5) * (6/7) * (12/11) * (18/19) * ... = 2/sqrt(3). - Dimitris Valianatos, Mar 27 2017
Primes p such that Kronecker(12,p) = +1 (12 is the discriminant of Q[sqrt(3)]), that is, odd primes that have 3 as a quadratic residue. - Jianing Song, Nov 21 2018
Comment from Richard R. Forberg, Feb 07 2023: (Start)
Conjecture: These are the exclusive prime factors of the set of integers d > 1 such that there exist primitive Heronian triangles with sides {b, b+d, b+2d} for one or more integers b.
Also b is always > d. For d=11 the b values begin {15, 17, 65, 75, 267, 305, 1025, ...}. For d=1 (not prime, thus not listed) the b values are given by A016064. (End)

			For p = 5, 3^2 - 1 = 8 <> 3*k for any integer k, so 5 is not in this sequence.
For p = 11, 3^5 - 1 = 242 = 11*22, so 11 is in this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to sequences related to decomposition of primes in quadratic fields

Cf. A003630, A010051, A038874, A091998.

a097933 n = a097933_list !! (n-1)
a097933_list = [x | x <- a091998_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

[p: p in PrimesUpTo(1000) | p mod 24 in [1, 11, 13, 23]]; // Vincenzo Librandi, Mar 23 2013

Select[Prime[Range[300]], MemberQ[{1, 11, 13, 23}, Mod[#, 24]]&] (* Vincenzo Librandi, Mar 23 2013 *)
Select[Prime[Range[2,200]],PowerMod[3,(#-1)/2,#]==1&] (* Harvey P. Dale, Jun 02 2020 *)

/* s = +-1, d=diff */ ptopm1d2(n,x,d,s) = { forprime(p=3,n,p2=(p-1)/2; y=x^p2 + s*(x-d)^p2; if(y%p==0,print1(p","))) }

{a(n)= local(m, c); if(n<1, 0, c=0; m=0; while( cMichael Somos, Aug 28 2006 */

A025021 Numbers whose least quadratic nonresidue (A020649) is 3.

Original entry on oeis.org

7, 14, 17, 31, 34, 41, 49, 62, 79, 82, 89, 98, 103, 113, 119, 127, 137, 151, 158, 161, 178, 199, 206, 217, 223, 226, 233, 238, 254, 257, 271, 274, 281, 287, 289, 302, 322, 329, 343, 353, 367, 391, 398, 401, 434, 439, 446, 449, 463, 466, 487, 497, 511, 514, 521, 527, 542

Offset: 1

David W. Wilson

nonn

n such that n is not divisible by 4, all primes dividing n are in A038873, and at least one prime dividing n is in A003630. - Robert Israel, Jul 19 2017

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Quadratic Nonresidue.

Cf. A020649, A025020, A025022, A025023, A025024, A025025, A025026, A025027, A025028, A025029.

Cf. A003630, A038873.

select(t -> numtheory:-quadres(2,t) = 1 and numtheory:-quadres(3,t)=-1, [$1..1000]); # Robert Israel, Jul 19 2017

Select[Range[500], Min @ Complement[Range[# - 1], Mod[Range[#/2]^2, #]] == 3 &] (* Amiram Eldar, Oct 31 2020 *)

residue(n,m)={local(r);r=0;for(i=1,floor(m/2),if(i^2%m==n,r=1));r}
isA025021(n)=residue(2,n) && !residue(3,n) \\ Michael B. Porter, Apr 18 2010

A038875 Primes p with legendre(3,p) = -1.

Original entry on oeis.org

2, 5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, 199, 211, 223, 233, 257, 269, 271, 281, 283, 293, 307, 317, 331, 353, 367, 379, 389, 401, 439, 449, 461, 463, 487, 499, 509, 521, 523, 547, 557, 569, 571, 593

Offset: 1

N. J. A. Sloane

nonn

Apart from the first term, primes p such that 3 is not a square mod p.

Apart from the first term, identical to A003630.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A040117, A003630.

Select[Prime@Range[120], JacobiSymbol[3, #] == -1 &] (* Vincenzo Librandi, Sep 09 2012 *)

isok(p) = isprime(p) && (kronecker(3, p) == -1); \\ Michel Marcus, Jan 24 2023

Edited by D. S. McNeil, R. J. Mathar and N. J. A. Sloane, Aug 15 2010

A038877 Primes p such that 6 is not a square mod p.

Original entry on oeis.org

7, 11, 13, 17, 31, 37, 41, 59, 61, 79, 83, 89, 103, 107, 109, 113, 127, 131, 137, 151, 157, 179, 181, 199, 223, 227, 229, 233, 251, 257, 271, 277, 281, 347, 349, 353, 367, 373, 397, 401, 419, 421, 439, 443, 449

Offset: 1

N. J. A. Sloane

Contribution from Cino Hilliard, Sep 06 2004: (Start)
Also primes p such that p divides 3^(p-1)/2 + 2^(p-1)/2.
Also primes p such that p divides 6^(p-1)/2 + 1.
Also primes p such that p divides 6^(p-1)/2 + 4^(p-1)/2. (End)
Inert rational primes in the field Q(sqrt(6)). - Alonso del Arte, Oct 14 2012
Primes congruent to 7, 11, 13, or 17 mod 24.

			17 is in the sequence because there is no solution to the equation x^2 - 6y = 17 in integers.
19 is NOT in the sequence because x^2 - 6y = 19 has solutions in integers, as does x^2 - 6y^2 = 19, e.g., x = 5, y = 1, and therefore (5 - sqrt(6))*(5 + sqrt(6)) = 19.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index to sequences related to decomposition of primes in quadratic fields

Cf. A003630.

Select[Prime@Range[120], JacobiSymbol[6, #] == -1 &] (* Vincenzo Librandi, Sep 08 2012 *)

forprime(p=2,500,if(kronecker(6,p)==-1, print1(p,", ")));
/* Joerg Arndt, Oct 15 2012 */

a(n) ~ 2n log n. - Charles R Greathouse IV, Oct 15 2012

Offset changed from 0 to 1 by Vincenzo Librandi, Sep 08 2012

A258328 L.g.f.: log(1 + Sum_{n>=1} x^(n^2) + x^(3*n^2) ).

Original entry on oeis.org

1, -1, 4, -1, 1, -4, 1, -1, 13, -11, 12, -16, 14, -15, 19, -1, 1, -13, 1, -11, 25, -12, 24, -40, 26, -14, 40, -15, 1, -29, 1, -1, 48, -35, 36, -61, 38, -39, 56, -11, 1, -39, 1, -12, 73, -24, 48, -88, 50, -36, 55, -14, 1, -40, 12, -15, 61, -59, 60, -101, 62, -63, 97, -1, 14, -48, 1, -35, 96, -60, 72, -157, 74, -38, 119, -39, 12, -56, 1, -11, 121, -83, 84, -135, 86, -87, 91, -12, 1, -83, 14, -24, 97, -48, 96, -184, 98, -64, 156, -36, 1, -89, 1, -14, 180, -107, 108, -196, 110, -132, 152, -15, 1, -99, 24, -59, 182, -60, 120, -245, 133

Offset: 1

Paul D. Hanna, Jun 03 2015

sign

			L.g.f.: L(x) = x - x^2/2 + 4*x^3/3 - x^4/4 + x^5/5 - 4*x^6/6 + x^7/7 - x^8/8 + 13*x^9/9 - 11*x^10/10 + 12*x^11/11 - 16*x^12/12 + 14*x^13/13 - 15*x^14/14 + 19*x^15/15 - x^16/16 +...+ a(n)*x^n/n +...
where
exp(L(x)) = 1 + x + x^3 + x^4 + x^9 + x^12 + x^16 + x^25 + x^27 + x^36 + x^48 + x^49 + x^64 + x^75 + x^81 + x^100 + x^108 +...+ x^(n^2) + x^(3*n^2) +...
Note that for n>1, a(n) = +1 at positions:
[5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, 101, 103, 113, 127, ...];
which appears to be A003630 (primes p such that 3 is not a square mod p).

Paul D. Hanna, Table of n, a(n) for n = 1..1024

Cf. A256357, A003630, A038875.

{a(n) = local(L=x); L = log(1 + sum(k=1,sqrtint(n+1), x^(k^2) + x^(3*k^2)) +x*O(x^n)); n*polcoeff(L,n)}
for(n=1,121, print1(a(n),", "))

a(n) = -1 iff n = 2^k for k>=1 [conjecture].

a(p) = +1 for primes p such that 3 is not a square mod p (A003630), and a(n) = +1 nowhere else except at n=0 [conjecture].

A296933 Primes p such that Legendre(3,p) = 0 or 1.

Original entry on oeis.org

3, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577

Offset: 1

N. J. A. Sloane, Dec 26 2017

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000

This is A038874 without the initial 2.

Cf. A003630, A038875, A097933, A296920.

# Load the Maple program HH given in A296920. Then run HH(3, 200); This produces A097933, A003630, this sequence, and A038875.

Join[{3}, Select[Prime[Range[200]], JacobiSymbol[3, #] == 1 &]] (* Paolo Xausa, May 11 2024 *)

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A035194 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 12.

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A097933 Primes p that divide 3^((p-1)/2) - 1.

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A025021 Numbers whose least quadratic nonresidue (A020649) is 3.

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A038875 Primes p with legendre(3,p) = -1.

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A038877 Primes p such that 6 is not a square mod p.

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A258328 L.g.f.: log(1 + Sum_{n>=1} x^(n^2) + x^(3*n^2) ).

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A296933 Primes p such that Legendre(3,p) = 0 or 1.

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A035194 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 12.