%I #14 Jun 12 2021 08:50:53

%S 20,42,100,110,156,272,294,342,500,506,812,930,1210,1332,1640,1806,

%T 2028,2058,2162,2500,2756,3422,3660,4422,4624,4970,5256,6162,6498,

%U 6806,7832,9312,10100,10506,11026,11342,11638,11772,12500,12656,13310,14406,16002,17030

%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 m-th alternating generalized harmonic number H'(m,k), for k = 2.

%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 geometric progressions of the form (p-1)*p^k for k > 0 and some primes p > 3. Note the factorization of initial terms of {a(n)} = {4*5, 6*7, 4*5^2, 10*11, 12*13, 16*17, 6*7^2, 18*19, 4*5^3, 22*23, 28*29, 30*31, 10*11^2, 36*37, 40*41, 42*43, 12*13^2, 6*7^3, 46*47, 4*5^4, 52*53, 58*59, 60*61, 66*67, 16*17^2, 70*71, 72*73, 78*79, 18*19^2, 82*83, ...}. The smallest term that does not fit this pattern is 11026 = ((149-1)/2) * 149.

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

%t k=2; 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,1,7000} ]

%Y Similar sequences for generalized harmonic numbers with different k: A125581 (k=1), A128673 (k=3), A128674 (k=4), A128675 (k=5); A128676 (k=6).

%Y For the least numbers k > 0 such that k^n does not divide the denominator of H(k,n) nor the denominator of H'(k,n), see A128670. See also A128671(n) = A128670(prime(n)).

%Y Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A125581, A126196, A126197, A128674, A128675, A128676, A128673, A128670, A129671.

%K nonn

%O 1,1

%A _Alexander Adamchuk_, Mar 20 2007

%E Edited and extended by _Max Alekseyev_, May 07 2010