cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A097956 Primes p such that p divides 5^(p-1)/2 - 3^(p-1)/2.

Original entry on oeis.org

7, 11, 17, 43, 53, 59, 61, 67, 71, 103, 109, 113, 127, 131, 137, 163, 173, 179, 181, 191, 197, 223, 229, 233, 239, 241, 251, 257, 283, 293, 307, 311, 317, 349, 353, 359, 367, 409, 419, 421, 431, 463, 479, 487, 491, 523, 541, 547, 557, 593, 599, 601, 607, 617
Offset: 1

Views

Author

Cino Hilliard, Sep 06 2004

Keywords

Comments

From Jianing Song, Oct 13 2022: (Start)
Rational primes that decompose in the field Q(sqrt(15)).
Primes p such that kronecker(60,p) = 1.
Primes congruent to 1, 7, 11, 17, 43, 49, 53, 59 modulo 60. (End)

Examples

			7 is a term since 5^3 - 3^3 = 7*14.

Links

Crossrefs

A038887, the sequence of primes that do not remain inert in the field Q(sqrt(15)), is essentially the same.
Cf. A038888 (rational primes that remain inert in the field Q(sqrt(15))).

Programs

  • Mathematica
    Select[Prime[Range[150]],Divisible[5^((#-1)/2)-3^((#-1)/2),#]&] (* Harvey P. Dale, Apr 11 2018 *)
  • PARI
    \\ 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, 5, 2, -1)
  • PARI
    isA097956(p) == isprime(p) && kronecker(60, p) == 1 \\ Jianing Song, Oct 13 2022

A035197 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 15.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 2, 3, 0, 4, 1, 5, 2, 2, 0, 3, 2, 4, 0, 4, 1, 0, 1, 6, 0, 2, 0, 6, 2, 4, 2, 3, 0, 0, 0, 4, 0, 4, 2, 6, 1, 0, 0, 5, 3, 2, 2, 0, 2, 2, 2, 8, 0, 0, 2, 3, 2, 0, 2, 7, 0, 4, 2, 6, 0, 4, 2, 4, 0, 0, 1, 0, 4, 0, 0, 5, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A038887, A038888.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[15, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
  • PARI
    my(m=15); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(15, d)); \\ Amiram Eldar, Nov 18 2023

Formula

From Amiram Eldar, Nov 18 2023: (Start)
a(n) = Sum_{d|n} Kronecker(15, d).
Multiplicative with a(p^e) = 1 if Kronecker(15, p) = 0 (p = 3 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(15, p) = -1 (p is in A038888), and a(p^e) = e+1 if Kronecker(15, p) = 1 (p is in A038887 \ {3, 5}).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*log(sqrt(15)+4)/sqrt(15) = 2.131108641007... . (End)

A347816 Prime numbers p such that both 15 and 85 are quadratic nonresidue (mod p).

Original entry on oeis.org

13, 29, 31, 41, 47, 79, 83, 139, 157, 199, 211, 263, 269, 373, 379, 383, 401, 433, 439, 443, 449, 457, 467, 499, 521, 563, 571, 577, 587, 613, 619, 641, 647, 691, 733, 751, 757, 809, 811, 821, 863, 881, 929, 937, 941, 991, 1033, 1049, 1051, 1061
Offset: 1

Views

Author

Sela Fried, Sep 15 2021

Keywords

Comments

Primes p such that E_6(x)/(x + 1) is irreducible (mod p) where E_6(x) is the Eulerian polynomial and E_6(x)/(x + 1) = x^4 + 56x^3 + 246x^2 + 56x + 1. (See A159041.)
The sequence is infinite.
It is the intersection of A038888 and A038972.

Links

Crossrefs

Cf. A159041, A008292, A173018, A038888, A038972, A347815.

Programs

  • Maple
    alias(ls = NumberTheory:-LegendreSymbol):
isA347816 := k -> isprime(k) and ls(15, k) = -1 and ls(85, k) = -1:
A347816List := upto -> select(isA347816, [`$`(3..upto)]):
A347816List(1061); # Peter Luschny, Sep 16 2021
  • Mathematica
    Select[Prime@Range[180], JacobiSymbol[15, #] == -1 && JacobiSymbol[85,#]==-1 &] (* Stefano Spezia, Sep 16 2021 *)
  • PARI
    isok(p) = isprime(p) && (kronecker(15,p)==-1) && (kronecker(85,p)==-1); \\ Michel Marcus, Sep 16 2021
  • Python
    from sympy.ntheory import legendre_symbol, primerange
A347816_list = [p for p in primerange(3,10**5) if legendre_symbol(15,p) == legendre_symbol(85,p) == -1] # Chai Wah Wu, Sep 16 2021
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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