A128674 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 = 4.

42, 110, 156, 272, 294, 342, 506, 812, 930, 1210, 1332, 1640, 1806, 2028, 2058, 2162, 2756, 3422, 3660, 4422, 4624, 4970, 5256, 6162, 6498, 6806, 7832, 9312, 10100, 10506, 11342, 11638, 11772, 12656, 13310, 14406, 16002, 17030, 18632, 19182, 22052, 22650, 23548, 24492, 26364

Offset: 1

Author

Alexander Adamchuk, Mar 20 2007

Keywords

• nonn

Comments

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.

Sequence contains geometric progressions of the form (p-1)*p^k for k > 0 and some prime p > 5. Note the factorization of initial terms of {a(n)} = {6*7, 10*11, 12*13, 16*17, 6*7^2, 18*19, 22*23, 28*29, 30*31, 10*11*2, 36*37, 40*41, 42*43, 12*13^2, 6*7^3, ...}. See more details in Comments at A128672 and A125581.

Links

• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A125581, A126196, A126197, A128672, A128675, A128676.

Programs

Mathematica

k=4; 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} ]

Extensions

Edited and extended by Max Alekseyev, May 09 2010