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 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. 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified July 24 20:26 EDT 2021. Contains 346273 sequences. (Running on oeis4.)