A038875 -id:A038875 - cp's OEIS frontend

A091338 a(n) = (3/n), where (k/n) is the Kronecker symbol.

1, -1, 0, 1, -1, 0, -1, -1, 0, 1, 1, 0, 1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, -1, 0, -1, -1, 0, -1, -1, 0, 1, 1, 0, 1, 1, 0, 1, -1, 0, -1, 1, 0, -1, 1, 0, 1, -1, 0, 1, -1, 0, -1, 1, 0, 1, 1, 0, 1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, -1, 0, -1, -1, 0, -1, -1, 0, 1, 1, 0, 1, 1, 0, -1, -1, 0, -1, 1, 0, -1, 1, 0, 1, -1, 0, 1, -1, 0

Offset: 1

Eric W. Weisstein, Dec 30 2003

a(2n+1) has period 6, i.e., if n == 1 (mod 2) then a(n+12) = a(n). A.H.M. Smeets, Jan 23 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jean-Paul Allouche, Leo Goldmakher, Mock characters and the Kronecker symbol, arXiv:1608.03957 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Kronecker Symbol

[KroneckerSymbol(3,n): n in [1..100]]; // Vincenzo Librandi, Aug 16 2016

A091338 := proc(n)
        numtheory[jacobi](3,n) ;
end proc: # R. J. Mathar, Nov 03 2011

Table[KroneckerSymbol[3, n], {n, 1, 100}] (* Vincenzo Librandi, Aug 16 2016 *)

PARI
```
a(n)=kronecker(3,n)
```

If n==0 (mod 3) a(n)=0; for p ==1 or 11 (mod 12) (i.e., p>3 in A038874), a(p)=+1; for p==2, 5 or 7 (mod 12) (i.e., p in A038875), a(p)=-1. - Benoit Cloitre, Jan 03 2004
From A.H.M. Smeets, Aug 01 2018: (Start)
Conjecture:
a(n) = 0 if and only if (n mod 3 = 0),
a(n) = 1 if (n mod 12 = 1 or n mod 12 = 11 or n mod 48 = 4 or n mod 48 = 44),
a(n) = -1 if (n mod 12 = 5 or n mod 12 = 7 or n mod 48 = 20 or n mod 48 = 28),
a(2) = -1, a(12*n+10) = -a(12*n+2) and a(12*n+14) = a(12*n+10) for n >= 0,
a(24*n+8) = -a(12*n+4) and a(24*n+16) = -a(12*n+4) for n >= 0. (End)
From A.H.M. Smeets, Aug 01 2018: (Start)
a(2*n+1) = 1 if and only if (n mod 6 = 0 or n mod 6 = 5),
a(2*n+1) = -1 if and only if (n mod 6 = 2 or n mod 6 = 3),
a(2*n+1) = 0 if and only if n mod 3 = 1,
a(2*n) = -a(n). (End)

More terms from Benoit Cloitre, Jan 03 2004

A035186 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 3.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

Cf. A038875, A097933.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[3, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

my(m=3); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(3, d)); \\ Amiram Eldar, Nov 20 2023

From Amiram Eldar, Oct 17 2022: (Start)
a(n) = Sum_{d|n} Kronecker(3, d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(2+sqrt(3)) / (3*sqrt(3)) = 0.506897... . (End)
Multiplicative with a(3^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(3, p) = -1 (p is in A038875), and a(p^e) = e+1 if Kronecker(3, p) = 1 (p is in A097933). - Amiram Eldar, Nov 20 2023

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 *)

cp's OEIS Frontend

A091338 a(n) = (3/n), where (k/n) is the Kronecker symbol.

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A035186 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 3.

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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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A035186 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 3.