cp's OEIS Frontend

A128676 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 = 6.

%I A128676 #14 Feb 16 2025 08:33:05
%S A128676 20,100,110,156,161,272,342,345,500,506,812,930,1210,1332,1640,1806,
%T A128676 2028,2162,2500,2756,3051,3422,3660,3703,4422,4624,4970,5256,6162,
%U A128676 6498,6806,7832,7935,9312,9605,10100,10506,11342,11638,11772,12500,12656,13310
%N A128676 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 = 6.
%C A128676 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 A128676 Sequence contains all terms of geometric progressions of the form (p-1)*p^k, k > 0, for some primes p >= 5, such as 4*5^k, 7*23^k, 15*23^k, 27*113^k, etc. Note the factorization of initial terms of {a(n)} = {4*5, 4*5^2, 10*11, 12*13, 7*23, 16*17, 18*19, 15*23, 4*5^3, 22*23, 28*29, 30*31, 10*11^2, 36*37, 40*41, 42*43, 12*13^2, 46*47, 4*5^4, 52*53, 27*113, 58*59, 60*61, 7*23^2, ...}. See more details in Comments at A128672 and A125581.
%H A128676 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%t A128676 k=6; 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,1,3703} ]
%Y A128676 Cf. A001008, A002805, A058313, A058312.
%Y A128676 Cf. A007406, A007407, A119682, A007410, A120296, A099828, A125581, A126196, A126197, A128672, A128673, A128676.
%K A128676 nonn
%O A128676 1,1
%A A128676 _Alexander Adamchuk_, Mar 20 2007
%E A128676 Edited and extended by Max Alekseyev, May 08 2010