A038880 -id:A038880 - cp's OEIS frontend

A035192 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 10.

1, 1, 2, 1, 1, 2, 0, 1, 3, 1, 0, 2, 2, 0, 2, 1, 0, 3, 0, 1, 0, 0, 0, 2, 1, 2, 4, 0, 0, 2, 2, 1, 0, 0, 0, 3, 2, 0, 4, 1, 2, 0, 2, 0, 3, 0, 0, 2, 1, 1, 0, 2, 2, 4, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 2, 0, 2, 0, 0, 0, 2, 3, 0, 2, 2, 0, 0, 4, 2, 1, 5

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 40. 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. A038880, A097955.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[10, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)

my(m=10); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(10, d)); \\ Amiram Eldar, Nov 18 2023

From Amiram Eldar, Nov 18 2023: (Start)
a(n) = Sum_{d|n} Kronecker(10, d).
Multiplicative with a(p^e) = 1 if Kronecker(10, p) = 0 (p = 2 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(10, p) = -1 (p is in A038880), and a(p^e) = e+1 if Kronecker(10, p) = 1 (p is in A097955).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(sqrt(10)+3)/sqrt(10) = 1.1500865228... . (End)

A377174 Primes p such that 5/2 is a primitive root modulo p.

Original entry on oeis.org

11, 17, 23, 47, 59, 73, 101, 103, 109, 113, 137, 139, 149, 167, 179, 211, 223, 229, 233, 257, 263, 269, 313, 337, 349, 353, 367, 379, 383, 389, 419, 421, 433, 461, 487, 499, 503, 509, 593, 607, 617, 647, 659, 661, 673, 727, 743, 811, 821, 823, 829, 857, 859, 863, 887, 941, 953, 967, 971, 977, 983

Offset: 1

Jianing Song, Oct 18 2024

If p is a term in this sequence, then 5/2 is not a square modulo p (i.e., p is in A038880).

Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Wikipedia, Artin's conjecture on primitive roots.
Index entries for primes by primitive root

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), this sequence (a=5), A377176 (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), A377177 (a=7), A377179 (a=9).
Cf. A038880, A005596.

forprime(p=7, 10^3, if(znorder(Mod(5/2, p))==p-1, print1(p, ", ")));

A007348 Primes for which -10 is a primitive root.

Original entry on oeis.org

3, 17, 29, 31, 43, 61, 67, 71, 83, 97, 107, 109, 113, 149, 151, 163, 181, 191, 193, 199, 227, 229, 233, 257, 269, 283, 307, 311, 313, 337, 347, 359, 389, 431, 433, 439, 443, 461, 467, 479, 509, 523, 541, 563, 577, 587, 593, 599, 631, 683, 701, 709, 719, 787, 821, 827, 839

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

Union of long period primes (A006883) of the form 4k+1 and half period primes (A097443) of the form 4k+3. - Davide Rotondo, Aug 25 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 24.8, p. 864.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993
Index entries for primes by primitive root

Cf. A038880.

Cf. A006883, A097443.

pr=-10; Select[Prime[Range[200 ] ], MultiplicativeOrder[pr, # ] == #-1 & ]

is(n)=gcd(n,10)==1 && znorder(Mod(-10,n))==n-1 \\ Charles R Greathouse IV, Nov 25 2014

More terms from N. J. A. Sloane, Apr 24 2005
Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar
A&S reference and Mathematica program corrected by T. D. Noe, Nov 04 2009

A097955 Primes p such that p divides 5^((p-1)/2) - 2^((p-1)/2).

Original entry on oeis.org

3, 13, 31, 37, 41, 43, 53, 67, 71, 79, 83, 89, 107, 151, 157, 163, 173, 191, 197, 199, 227, 239, 241, 271, 277, 281, 283, 293, 307, 311, 317, 347, 359, 373, 397, 401, 409, 431, 439, 443, 449, 467, 479, 521, 523, 547, 557, 563, 569, 587, 599, 601, 613, 631, 641

Offset: 1

Cino Hilliard, Sep 06 2004

Also 3 and primes p such that (p^2 - 1)/24 mod 10 = {0, 7}. - Richard R. Forberg, Aug 31 2013
Also primes p such that x^2 = 10 mod p has integer solutions, or Legendre(10, p) = 1. However, p could be irreducible but not prime in Z[sqrt(10)], especially if p = 3 or 7 mod 10. - Alonso del Arte, Dec 27 2015
Rational primes that decompose in the field Q(sqrt(10)). - N. J. A. Sloane, Dec 26 2017
From Jianing Song, Oct 13 2022: (Start)
Primes p such that kronecker(10,p) = 1 (or equivalently, kronecker(40,p) = 1).
Primes congruent to 1, 3, 9, 13, 27, 31, 37, 39 modulo 40. (End)

			For p = 13, 5^6 - 2^6 = 15561 is divisible by 13, so 13 is in the sequence.

A038879, the sequence of primes that do not remain inert in the field Q(sqrt(10)), is essentially the same.

Cf. A038880 (rational primes that remain inert in the field Q(sqrt(10))).

select(p -> isprime(p) and  10 &^ ((p-1)/2)  mod p = 1, [seq(i,i=3..1000,2)]); # Robert Israel, Dec 28 2015

Select[Prime[Range[100]], JacobiSymbol[10, #] == 1 &] (* Alonso del Arte, Dec 27 2015 *)

\\ 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, 3, -1)

isA097955(p) == isprime(p) && kronecker(10,p) == 1 \\ Jianing Song, Oct 13 2022

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A035192 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 10.

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A377174 Primes p such that 5/2 is a primitive root modulo p.

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A007348 Primes for which -10 is a primitive root.

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A097955 Primes p such that p divides 5^((p-1)/2) - 2^((p-1)/2).

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A035192 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 10.