A128672 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.

20, 42, 100, 110, 156, 272, 294, 342, 500, 506, 812, 930, 1210, 1332, 1640, 1806, 2028, 2058, 2162, 2500, 2756, 3422, 3660, 4422, 4624, 4970, 5256, 6162, 6498, 6806, 7832, 9312, 10100, 10506, 11026, 11342, 11638, 11772, 12500, 12656, 13310, 14406, 16002, 17030

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 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.

Links

• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

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

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)).

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

Programs

Mathematica

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} ]

Extensions

Edited and extended by Max Alekseyev, May 07 2010