A038904 -id:A038904 - cp's OEIS frontend

A035212 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 30.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 0, 0, 1, 1, 2, 1, 2, 2, 1, 0, 1, 0, 2, 2, 1, 2, 2, 2, 1, 0, 2, 0, 0, 1, 0, 0, 1, 3, 1, 2, 2, 0, 1, 0, 2, 2, 2, 0, 1, 0, 0, 2, 1, 2, 0, 0, 2, 0, 2, 2, 1, 0, 2, 1, 2, 0, 2, 0, 1, 1

Offset: 1

N. J. A. Sloane

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A038904, A097959.

a[n_] := DivisorSum[n, KroneckerSymbol[30, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)

my(m = 30); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(30, d)); \\ Amiram Eldar, Nov 19 2023

From Amiram Eldar, Nov 19 2023: (Start)
a(n) = Sum_{d|n} Kronecker(30, d).
Multiplicative with a(p^e) = 1 if Kronecker(30, p) = 0 (p = 2, 3 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(30, p) = -1 (p is in A038904), and a(p^e) = e+1 if Kronecker(30, p) = 1 (p is in A097959).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(2*sqrt(30)+11)/sqrt(30) = 1.127932330891... . (End)

A097959 Primes p such that p divides 6^((p-1)/2) - 5^((p-1)/2).

Original entry on oeis.org

7, 13, 17, 19, 29, 37, 71, 83, 101, 103, 107, 113, 127, 137, 139, 149, 157, 191, 211, 223, 227, 233, 239, 241, 257, 269, 277, 311, 331, 347, 353, 359, 367, 373, 379, 389, 397, 409, 431, 443, 461, 463, 467, 479, 487, 499, 509, 563, 571, 587, 593, 599, 601, 607

Offset: 1

Cino Hilliard, Sep 06 2004

From Jianing Song, Oct 13 2022: (Start)
Rational primes that decompose in the field Q(sqrt(30)).
Primes p such that kronecker(30,p) = 1 (or equivalently, kronecker(120,p) = 1).
Primes congruent to 1, 7, 13, 17, 19, 29, 37, 49, 71, 83, 91, 101, 103, 107, 113, 119 modulo 120. (End)

			7 is a term since it is a prime and 6^((7-1)/2) - 5^((7-1)/2) = 6^3 - 5^3 = 91 = 7*13 is divisible by 7.

A038903, the sequence of primes that do not remain inert in the field Q(sqrt(30)), is essentially the same.

Cf. A038904 (rational primes that remain inert in the field Q(sqrt(30))).

Select[Prime[Range[200]],Divisible[6^((#-1)/2)-5^((#-1)/2),#]&] (* Harvey P. Dale, Jun 06 2018 *)
Select[Range[3, 600, 2], PrimeQ[#] && PowerMod[5, (# - 1)/2, #] == PowerMod[6, (# - 1)/2, #] &] (* Amiram Eldar, Apr 07 2021 *)

\\ 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","))) }
ptopm1d2(1000, 6, 1, -1)

isA097959(p) == isprime(p) && kronecker(30, p) == 1 \\ Jianing Song, Oct 13 2022

Definition clarified by Harvey P. Dale, Jun 06 2018

A347815 Prime numbers p such that both 30 and 105 are quadratic nonresidue (mod p).

Original entry on oeis.org

11, 31, 43, 47, 61, 67, 163, 167, 173, 179, 181, 193, 199, 229, 271, 281, 293, 337, 349, 383, 401, 439, 449, 457, 491, 503, 547, 569, 641, 647, 659, 661, 673, 677, 773, 797, 809, 829, 883, 887, 907, 983, 1013, 1019, 1021, 1033, 1039, 1069, 1223, 1231

Offset: 1

Sela Fried, Sep 15 2021

nonn

Primes p such that the Eulerian polynomial E_5(x) is irreducible (mod p), where E_5(x) = x^4 + 26x^3 + 66x^2 + 26x + 1.

The sequence is infinite.

A. J. J. Heidrich, On the factorization of Eulerian polynomials, Journal of Number Theory, 18(2):157-168, 1984.

Cf. A038904, A008292, A173018, A347816.

Select[Prime@Range[205], JacobiSymbol[30, #] == -1 && JacobiSymbol[105,#]==-1 &] (* Stefano Spezia, Sep 16 2021 *)

isok(p) = isprime(p) && (kronecker(30, p)==-1) && (kronecker(105, p)==-1); \\ Michel Marcus, Sep 16 2021

from sympy.ntheory import legendre_symbol, primerange
A347815_list = [p for p in primerange(3,10**5) if legendre_symbol(30,p) == legendre_symbol(105,p) == -1] # Chai Wah Wu, Sep 16 2021

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A035212 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 30.

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A097959 Primes p such that p divides 6^((p-1)/2) - 5^((p-1)/2).

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A347815 Prime numbers p such that both 30 and 105 are quadratic nonresidue (mod p).

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cp's OEIS Frontend

A035212 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 30.

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A097959 Primes p such that p divides 6^((p-1)/2) - 5^((p-1)/2).

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A347815 Prime numbers p such that both 30 and 105 are quadratic nonresidue (mod p).

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A035212 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 30.