A128673 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 = 3.

94556602, 141834903, 189113204, 283669806, 450820422

Offset

1,1

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.

Note that {a(n)} contains the following geometric progressions: ((16843-1)/3)*16843^m found by Max Alekseyev, ((16843-1)/2)*16843^m found by Max Alekseyev, ((16843-1)*2/3)*16843^m, (16843-1)*16843^m, 20826*21647^m found by Max Alekseyev, ((2124679-1)/3)*2124679^m, ((2124679-1)/2)*2124679^m, ((2124679-1)*2/3)*2124679^m, (2124679-1)*2124679^m. Here {16843, 2124679} = A088164 are the only two currently known Wolstenholme Primes: primes p such that {2p-1} choose {p-1} == 1 mod p^4. See more details in Comments at A128672 and A125581.

Links

Table of n, a(n) for n=1..5.

Eric Weisstein's World of Mathematics, Harmonic Number

Eric Weisstein's World of Mathematics, Wolstenholme Prime

Mathematica

k=3; 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, 450820422} ]

Crossrefs

Cf. A001008, A002805, A058313, A058312.

Cf. A007406, A007407, A119682, A007410, A120296, A099828.

Cf. A125581, A126196, A126197, A128672, A128674, A128675, A128676, A128670, A128671.

Cf. A088164 (Wolstenholme primes).

Sequence in context: A293244 A136634 A033625 * A028502 A114662 A250964

Adjacent sequences: A128670 A128671 A128672 * A128674 A128675 A128676

Keyword

nonn,hard,more

Author

Alexander Adamchuk, Apr 18 2007

Status

approved