1,1

Generalized harmonic numbers are defined as H(n,k) = Sum[ 1/i^k, {i,1,n} ]. Alternating generalized harmonic numbers are defined as H'(n,k) = Sum[ (-1)^(i+1)*1/i^k, {i,1,n} ]. 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 for A128672 and A125581.

Eric Weisstein, Link to a section of The World of Mathematics. Harmonic Number.

Eric Weisstein, Link to a section of The World of Mathematics. Wolstenholme Prime.

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} ]

Cf. A001008, A002805, A058313, A058312. Cf. A007406, A007407, A119682, A007410, A120296, A099828. Cf. A125581 = numbers n such that n does not divide the denominator of the n-th harmonic number nor the denominator of the n-th alternating harmonic number. Cf. A126196, A126197. Cf. A128672 = numbers n such that n^k does not divide the denominator of the n-th generalized harmonic number H(n, k) nor the denominator of the n-th alternating generalized harmonic number H'(n, k), for k = 2. Cf. A128674, A128675, A128676. Cf. A128670 = Least number k>0 such that k^n does not divide the denominator of generalized harmonic number H(k, n) nor the denominator of alternating generalized harmonic number H'(k, n). Cf. A128671 = A128670(Prime(n)). Cf. A088164 = Wolstenholme Primes.

Sequence in context: A147527 A136634 A033625 this_sequence A028502 A114662 A075010

Adjacent sequences: A128670 A128671 A128672 this_sequence A128674 A128675 A128676

hard,more,nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 18 2007