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} ]. Conjecture: a(n) contains all terms of geometric progressions 37^k*(37-1)/3, 37^k*(37-1)/2, 37^k*(37-1)*2/3, 37^k*(37-1) for k>0. Note the factorization of initial terms of a(n) = {37*12, 37*18, 37*24, 37*36, ...}. See more details in comments for A128672 and A125581.

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

k=5; 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, 2000} ]

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. A128673, A128676.

Sequence in context: A031699 A098254 A111496 this_sequence A043507 A098255 A028460

Adjacent sequences: A128672 A128673 A128674 this_sequence A128676 A128677 A128678

hard,more,nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Mar 20 2007