A073276 Irregular primes (A000928) with irregularity index one.

37, 59, 67, 101, 103, 131, 149, 233, 257, 263, 271, 283, 293, 307, 311, 347, 389, 401, 409, 421, 433, 461, 463, 523, 541, 557, 577, 593, 607, 613, 619, 653, 659, 677, 683, 727, 751, 757, 761, 773, 797, 811, 821, 827, 839, 877, 881, 887, 953, 971, 1061, 1091

Offset: 1

Author

Robert G. Wilson v, Jul 22 2002

Keywords

• nonn

Comments

A prime p is regular if and only if the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) are not divisible by p.

In other words, irregular primes p dividing the numerator of B(2k) for a single k, 1<=k<(p-1)/2.

Links

• T. D. Noe, Table of n, a(n) for n=1..10000 (from Buhler et al.)
• J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. A. Shokrollahi, Irregular Primes and Cyclotomic Invariants to 12 Million, J. Symbolic Computation 31, 2001, 89-96.
• Bernoulli numbers, irregularity index of primes

Crossrefs

Cf. A000928, A000367, A060974, A060975 and A073277.

Programs

Mathematica

Do[p = Prime[n]; k = 1; c = 0; While[ 2*k < p - 3, If[ Mod[ Numerator[ BernoulliB[2*k]], p] == 0, c++ ]; k++ ]; If[ c == 1, Print[p]], {n, 3, 200} ]