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A069040 Numbers k that divide the numerator of B(2k) (the Bernoulli numbers).

%I A069040 #33 Apr 24 2025 02:14:46
%S A069040 1,5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,59,61,65,67,71,73,
%T A069040 77,79,83,85,89,91,95,97,101,103,107,109,113,115,119,121,125,127,131,
%U A069040 133,137,139,143,145,149,151,155,157,161,163,167,169,173,175,179,181
%N A069040 Numbers k that divide the numerator of B(2k) (the Bernoulli numbers).
%C A069040 Equivalently, k is relatively prime to the denominator of B(2k). Equivalently, there are no primes p such that p divides k and p-1 divides 2k. These equivalences follow from the von Staudt-Clausen and Sylvester-Lipschitz theorems.
%C A069040 The listed terms are the same as those in A070191, but the sequences are not identical. (The similarity is mostly explained by the absence of multiples of 2, 3 and 55 from both sequences.) See A070192 and A070193 for the differences.
%D A069040 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954.
%H A069040 Amiram Eldar, <a href="/A069040/b069040.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..300 from Seiichi Manyama)
%H A069040 I. Sh. Slavutskii, <a href="https://doi.org/10.1006/jnth.1995.1093">A note on Bernoulli numbers</a>, Journal of Number Theory, Vol. 53, No. 2 (1995), pp. 309-310.
%p A069040 A069040 := proc(n)
%p A069040     option remember;
%p A069040     if n=1 then
%p A069040         1;
%p A069040     else
%p A069040         for k from procname(n-1)+1 do
%p A069040             if numer(bernoulli(2*k)) mod k = 0 then
%p A069040                 return k;
%p A069040             end if;
%p A069040         end do:
%p A069040     end if;
%p A069040 end proc: # _R. J. Mathar_, Jan 06 2013
%t A069040 testb[n_] := Select[First/@FactorInteger[n], Mod[2n, #-1]==0&]=={}; Select[Range[200], testb]
%o A069040 (PARI) isok(k) = {my(p = factor(k)[,1]); for(i = 1, #p, if(!((2*k) % (p[i]-1)), return(0))); 1;} \\ _Amiram Eldar_, Apr 24 2025
%Y A069040 Cf. A027641, A070191, A070192, A070193.
%K A069040 nonn
%O A069040 1,2
%A A069040 _Benoit Cloitre_, Apr 03 2002
%E A069040 More information from _Dean Hickerson_, Apr 26 2002