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A128673 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 3.

%I A128673 #19 Feb 16 2025 08:33:05
%S A128673 94556602,141834903,189113204,283669806,450820422
%N A128673 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 3.
%C A128673 Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.
%C A128673 Note that {a(n)} contains the following geometric progressions: ((16843-1)/3)*16843^m found by _Max Alekseyev_, ((16843-1)/2)*16843^m found by _Max Alekseyev_, ((16843-1)*2/3)*16843^m, (16843-1)*16843^m, 20826*21647^m found by _Max Alekseyev_, ((2124679-1)/3)*2124679^m, ((2124679-1)/2)*2124679^m, ((2124679-1)*2/3)*2124679^m, (2124679-1)*2124679^m. Here {16843, 2124679} = A088164 are the only two currently known Wolstenholme Primes: primes p such that {2p-1} choose {p-1} == 1 mod p^4. See more details in Comments at A128672 and A125581.
%H A128673 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%H A128673 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmePrime.html">Wolstenholme Prime</a>
%t A128673 k=3; f=0; g=0; Do[ f=f+1/n^k; g=g+(-1)^(n+1)*1/n^k; kf=Denominator[f]; kg=Denominator[g]; If[ !IntegerQ[kf/n^k] && !IntegerQ[kg/n^k], Print[n] ], {n, 1, 450820422} ]
%Y A128673 Cf. A001008, A002805, A058313, A058312.
%Y A128673 Cf. A007406, A007407, A119682, A007410, A120296, A099828.
%Y A128673 Cf. A125581, A126196, A126197, A128672, A128674, A128675, A128676, A128670, A128671.
%Y A128673 Cf. A088164 (Wolstenholme primes).
%K A128673 nonn,hard,more
%O A128673 1,1
%A A128673 _Alexander Adamchuk_, Apr 18 2007