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A076637 Numerators of harmonic numbers when these numerators are divisible by squares of primes >= 5 in the case of Wolstenholme's Theorem.

%I A076637 #35 Feb 16 2025 08:32:47
%S A076637 25,49,7381,86021,2436559,14274301,19093197,315404588903,
%T A076637 9304682830147,54801925434709,2078178381193813,12309312989335019,
%U A076637 5943339269060627227,14063600165435720745359,254381445831833111660789,15117092380124150817026911
%N A076637 Numerators of harmonic numbers when these numerators are divisible by squares of primes >= 5 in the case of Wolstenholme's Theorem.
%C A076637 By Wolstenholme's Theorem, if p prime >= 5, the numerator of the harmonic number H_{p-1} is always divisible by p^2. The obtained quotients are in A061002. - _Bernard Schott_, Dec 02 2018
%C A076637 The numbers 363, numerator of H_7 and 9227046511387, numerator of H_{29}, which have been found by Amiram Eldar and Michel Marcus, are also divisible by prime squares, respectively by 11^2 and 43^2, but not in the case of Wolstenholme's Theorem. So, a new sequence A322434 is created with all the numerators of Harmonic numbers which are divisible by any prime square >= 5. - _Bernard Schott_, Dec 08 2018
%H A076637 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>
%e A076637 25 is a term because the numerator of the harmonic number H_4 = 1 + 1/2+ 1/3 + 1/4 = 25/12 is divisible by the square of 5;
%e A076637 49 is a term because the numerator of the harmonic number H_6 = 1 + 1/2+ 1/3 + 1/4 + 1/5 + 1/6 = 49/20 is divisible by the square of 7.
%t A076637 a[p_] := Numerator[HarmonicNumber[p - 1]]; a /@ Prime@Range[3, 20] (* _Amiram Eldar_, Dec 08 2018 *)
%Y A076637 Cf. A076638, A001008.
%K A076637 nonn,frac
%O A076637 1,1
%A A076637 Michael Gilleland (megilleland(AT)yahoo.com), Oct 23 2002
%E A076637 More terms from _Amiram Eldar_, Dec 04 2018