A091888 Irregularity index of prime(n): number of numbers k, 1 <= k <= (p-3)/2, such that p = prime(n) divides the numerator of the Bernoulli number B(2k).

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

Offset: 2

Author

T. D. Noe and Benoit Cloitre, Feb 09 2004

Keywords

• nonn

Comments

Note offset is 2: only odd primes are considered.

Links

• Amiram Eldar, Table of n, a(n) for n = 2..10001

Crossrefs

Cf. A073277 (primes having irregularity index 2), A060975 (primes having irregularity index 3), A061576 (least prime having irregularity index n), A091887 (irregularity index of irregular prime A000928(n)).

Cf. A027641/A027642, A000367/A002445, A000928.

Programs

Mathematica

irregPrimeIndex[n_] := Block[{p = Prime[n], cnt = 0, k = 1}, While[ 2k + 2 < p, If[ Mod[ Numerator[ BernoulliB[ 2k]], p] == 0, cnt++]; k++]; cnt]; Array[ irregPrimeIndex, 105, 2] (* Robert G. Wilson v, Sep 20 2012 *)

PARI

a(n)=sum(i=1,(prime(n)-1)/2,if(numerator(bernfrac(2*i))%prime(n),0,1))  \\ corrected by Amiram Eldar, May 10 2022

Formula

0 if p is a regular prime; > 0 if p is an irregular prime.