A038878 -id:A038878 - cp's OEIS frontend

A089509 a(n) = Kronecker(7/n).

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

Offset: 1

Benoit Cloitre, Jan 03 2004

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.

Cf. A091338, A080891.

[KroneckerSymbol(7,n): n in [1..100]]; // Vincenzo Librandi, Aug 16 2016

Table[KroneckerSymbol[7, n], {n, 100}] (* Vincenzo Librandi, Aug 16 2016 *)

PARI
```
a(n)=kronecker(7,n)
```

if n==0 (mod 7) a(n)=0; for p in=A003632 a(p)=-1; for p in A038878 a(p)=+1

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A035189 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 7.

Original entry on oeis.org

1, 2, 2, 3, 0, 4, 1, 4, 3, 0, 0, 6, 0, 2, 0, 5, 0, 6, 2, 0, 2, 0, 0, 8, 1, 0, 4, 3, 2, 0, 2, 6, 0, 0, 0, 9, 2, 4, 0, 0, 0, 4, 0, 0, 0, 0, 2, 10, 1, 2, 0, 0, 2, 8, 0, 4, 4, 4, 2, 0, 0, 4, 3, 7, 0, 0, 0, 0, 0, 0, 0, 12, 0, 4, 2, 6, 0, 0, 0, 0, 5

Offset: 1

N. J. A. Sloane

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

Cf. A003632, A038878.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[7, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

my(m=7); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(7, d)); \\ Amiram Eldar, Nov 20 2023

From Amiram Eldar, Oct 17 2022: (Start)
a(n) = Sum_{d|n} Kronecker(7, d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(8+3*sqrt(7)) / sqrt(7) = 2.0929097... . (End)
Multiplicative with a(7^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(7, p) = -1 (p is in A003632), and a(p^e) = e+1 if Kronecker(7, p) = 1 (p is in A038878 \ {7}). - Amiram Eldar, Nov 20 2023

A296934 Rational primes that decompose in the field Q(sqrt(7)).

Original entry on oeis.org

3, 19, 29, 31, 37, 47, 53, 59, 83, 103, 109, 113, 131, 137, 139, 149, 167, 193, 197, 199, 223, 227, 233, 251, 271, 277, 281, 283, 307, 311, 317, 337, 367, 373, 383, 389, 401, 419, 421, 439, 449, 457, 467, 479, 503, 523, 541, 557, 563, 569, 587, 607, 613

Offset: 1

N. J. A. Sloane, Dec 26 2017

nonn

Load the Maple program HH given in A296920. Then run HH(5, 200); This produces A296934, A003632, A038878.

A247874 Lesser of twin primes (p, q=p+2) such that 7 is a square mod p and mod q.

Original entry on oeis.org

29, 137, 197, 281, 419, 617, 641, 809, 1061, 1091, 1229, 1289, 1427, 1481, 1877, 1931, 2129, 2237, 2267, 2381, 2549, 2657, 2687, 2801, 2969, 3329, 3359, 3389, 3527, 3557, 3581, 3917, 4001, 4229, 4337, 4421, 4481, 4649, 4787, 5009, 5657, 5741, 5849, 5879, 6131, 6269, 6299, 6551, 6689, 7307

Offset: 1

Zak Seidov, Sep 25 2014

nonn

Both p and p + 2 are terms in A038878.

All terms are congruent to {1, 25, 27} mod 28.

			7+29*1=36=6^2 and 7+31*3=100=10^2 hence 7 is a square mod 29 and mod 31.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A001359, A038878.

Select[Prime[Range[5,1000]], PrimeQ[# + 2] && JacobiSymbol[7, #] == JacobiSymbol[7, # + 2] == 1 &]

lista(nn) = {forprime(p=2, nn, if (isprime(q=p+2) && issquare(Mod(7, p)) && issquare(Mod(7, q)), print1(p, ", ")););} \\ Michel Marcus, Sep 25 2014

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A089509 a(n) = Kronecker(7/n).

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A035189 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 7.

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A296934 Rational primes that decompose in the field Q(sqrt(7)).

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A247874 Lesser of twin primes (p, q=p+2) such that 7 is a square mod p and mod q.

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A035189 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 7.