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

2,36

Comments

Note offset is 2: only odd primes are considered.

Links

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

Formula

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

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 *)

Prog

(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

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.

Sequence in context: A354449 A349436 A089811 * A379505 A379504 A083928

Adjacent sequences: A091885 A091886 A091887 * A091889 A091890 A091891

Keyword

nonn

Author

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

Status

approved