A092291 - OEIS

%I #14 Sep 28 2013 03:29:53

%S 574,1269,1910,3384,1185,1376,9611,4789,9670,20946,13019,11247,2689,

%T 22708,13355,45251,48407,32653,18761,38706,76391,25563,50310,79023,

%U 44948,29864,21716,71441,104339,22993,73572,61549,14714,26122,6227,179369,159687,5862,132157,24925,76023,15346,73479,136956,212240,10587,3801,137040,108520,194171,98550,282532,87272,133081,220187,305002,41764,27268,380180,70921,184940,241076,73858,80108,250927

%N Let p = n-th irregular prime, A000928(n). Then a(n) = smallest value of m such that numerator(Bernoulli(2*m)/(2*m)) / numerator(Bernoulli(2*m)/(2*m*(2*m-1))) equals p.

%C It was conjectured that a(n) = (1 + A000928(n) * (A035112(n) - 1))/2. However, Bernd Kellner's insightful paper shows that this formula first fails for the irregular prime 6449. - _T. D. Noe_, Feb 10 2004

%H Bernd Kellner, <a href="http://www.bernoulli.org/~bk/conjbn.pdf">A conjecture about numerators of Bernoulli numbers</a>

%t (* This program is not convenient for a large number of terms *) irregularPrimeQ[p_] := Module[{k = 1}, While[2*k <= p-3 && Mod[ Numerator[ BernoulliB[2*k]], p] != 0, k++]; 2*k <= p-3]; irregularPrime[1] = 37; irregularPrime[n_] := irregularPrime[n] = Module[{p}, For[p = NextPrime[ irregularPrime[n-1]], True, p = NextPrime[p], If[ irregularPrimeQ[p], Return[p]]]]; a[n_] := a[n] = For[m = 1, True, m++, If[ Numerator[BernoulliB[2*m]/(2*m)] / Numerator[ BernoulliB[2*m]/(2*m*(2*m-1))] == irregularPrime[n], Return[m]]]; Table[ Print[a[n]]; a[n], {n, 1, 15}] (* _Jean-François Alcover_, Sep 27 2013 *)

%Y Term in A090495 corresponding to first occurrence of p in A090496.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_, based on a suggestion of _Roland Bacher_, Feb 05 2004

%E Initial terms were computed by Roland Bacher, Feb 04 2004; further terms from _Hans Havermann_, Feb 05 2004 and _T. D. Noe_, Feb 06 2004

%E Offset modified by _Jean-François Alcover_, Sep 27 2013