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A093569
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For p = prime(n), the number of integers k < p-1 such that p divides A001008(k), the numerator of the harmonic number H(k).
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2
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0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 2, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| It is well-known that prime p >= 3 divides the numerator of H(p-1). For primes p in A092194, there are integers k < p-1 for which p divides the numerator of H(k). Interestingly, if p divides A001008(k) for k < p-1, then p divides A001008(p-k-1). Hence the terms of this sequence are usually even. The only exceptions are the two known Wieferich primes 1093 and 3511, A001220, which have 3 values of k < p-1 for which p divides A001008(k), one being k = (p-1)/2.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Wieferich Prime
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EXAMPLE
| a(5) = 2 because 11 = prime(5) and there are 2 values, k = 3 and 7, such that 11 divides A001008(k).
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MATHEMATICA
| len=500; Table[p=Prime[i]; cnt=0; k=1; While[k<p-1, If[Mod[Numerator[HarmonicNumber[k]], p]==0, cnt++ ]; k++ ]; cnt, {i, len}]
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CROSSREFS
| Cf. A001008, A001220, A092194.
Sequence in context: A000089 A051907 A178176 * A073091 A125250 A048113
Adjacent sequences: A093566 A093567 A093568 * A093570 A093571 A093572
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KEYWORD
| nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Apr 01 2004
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