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A128675
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 n-th alternating generalized harmonic number H'(m,k), for k = 5.
5
444, 666, 888, 1332, 16428, 24642, 32856, 49284, 607836, 911754, 1215672, 1823508
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.
Sequence contains all terms of geometric progressions 37^k*(37-1)/3, 37^k*(37-1)/2, 37^k*(37-1)*2/3, 37^k*(37-1) for k > 0. Note the factorization of initial terms of {a(n)} = {37*12, 37*18, 37*24, 37*36, ...}. See more details in Comments at A128672 and A125581.
LINKS
Eric Weisstein's World of Mathematics, Harmonic Number
MATHEMATICA
k=5; 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} ]
KEYWORD
hard,more,nonn
AUTHOR
Alexander Adamchuk, Mar 20 2007
EXTENSIONS
Eight more terms from Max Alekseyev, May 08 2010
STATUS
approved