A090798 Irregular primes in the ratio numerator(Bernoulli(2*n)/(2*n)) / numerator(Bernoulli(2*n)/(2*n*(2*n-r))) when these numerators are different and n is a minimum for some integer r. Duplication indicates irregularity index > 1.

37, 59, 67, 101, 103, 131, 149, 157, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 353, 379, 379, 389, 401, 409, 421, 433, 461, 463, 467, 467, 491, 491, 491, 523, 541, 547, 547, 557, 577, 587, 587, 593, 607, 613, 617, 617, 617, 619, 631, 631, 647

Offset

1,1

Comments

Only even values of r need to be tested.

See Table A.3, "Calculated irregular pairs of order 10 of primes below 1000," in B. C. Kellner.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..2000

Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007) 405-441.

Formula

Given a = numerator(Bernoulli(2*n)/(2*n)) and b = numerator(a/(2*n-r)) for integer r positive or negative, then n>0 n = p + r/2 For every irregular prime p there is an r such that n is minimum.

Mathematica

f[p_] := Block[{c = 0, k = 1}, While[ 2k <= p - 3, If[ Mod[ Numerator@ BernoulliB[ 2k], p] == 0, c++]; k++]; c]; p = 5; lst = {}; While[p < 1001, AppendTo[lst, Table[p, {f@ p}]]; p = NextPrime@ p]; Flatten@ lst

Prog

(PARI) \ prestore some ireg primes in iprime[] bernmin(m) = { for(x=1, m, p=iprime[x]; forstep(r=2, p, 2, n=r/2+p; n2=n+n; a = numerator(bernfrac(n2)/(n2)); \ A001067 b = numerator(a/(n2-r)); \ if(a <> b, print(r", "n", "a/b)) if(a <> b, print1(a/b", ")) ) ) }

Crossrefs

Cf. A090495 A090496.

Sequence in context: A179150 A127023 A109166 * A000928 A073276 A281290

Adjacent sequences: A090795 A090796 A090797 * A090799 A090800 A090801

Keyword

nonn

Author

Cino Hilliard, Feb 16 2004

Status

approved