A128675 - OEIS

%I #11 Jun 10 2021 22:31:05

%S 444,666,888,1332,16428,24642,32856,49284,607836,911754,1215672,

%T 1823508

%N 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 n-th alternating generalized harmonic number H'(m,k), for k = 5.

%C 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 Sequence 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 at A128672 and A125581.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>

%t 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} ]

%Y Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A099828, A125581, A126196, A126197, A128672, A128673, A128676.

%K hard,more,nonn

%O 1,1

%A _Alexander Adamchuk_, Mar 20 2007

%E Eight more terms from _Max Alekseyev_, May 08 2010