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A268112
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Numbers k for which the numerator of the k-th harmonic number H_k is divisible by the third power of a prime less than k.
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5
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OFFSET
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1,1
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COMMENTS
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The sequence contains numbers k for which there is a prime p < k with v_p(H_k) >= 3, where v_p(x) is the p-adic valuation of x and H_k is the k-th Harmonic number. All numbers were found by D. W. Boyd. The corresponding p for a(1) through a(4) is 11 while for a(5) (in the b-file) is 83. [Edited by Petros Hadjicostas, May 25 2020]
It is a widely believed conjecture that there is no pair of an integer k and a prime p for which v_p(H_k) >= 4. If variations of this conjecture hold, then Krattenhaler and Rivoal (2007-2009) would be able to establish some formulas for their theory. See also A007757, A131657, and A131658. - Petros Hadjicostas, May 25 2020
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LINKS
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PROG
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(PARI) h(n) = sum(i=1, n, 1/i);
is(n) = {forprime(p=1, n-1, if(valuation((numerator(h(n))), p) > 2, return(1))); return(0)} \\ Edited by Petros Hadjicostas, May 25 2020
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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