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A177783
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Wolstenholme quotient of prime p=A000040(n), i.e., such integer m<p that harmonic number H(p-1) == m*p^2 (mod p^3).
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3
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3, 6, 6, 7, 10, 14, 18, 20, 16, 24, 17, 38, 39, 19, 29, 28, 12, 53, 31, 19, 53, 58, 48, 42, 1, 33, 53, 37, 5, 81, 4, 17, 29, 13, 13, 72, 75, 70, 173, 159, 111, 150, 39, 178, 106, 163, 196, 163, 172, 30, 98, 24, 177, 261, 212, 223, 122, 147, 276, 17, 92, 111, 27, 209, 241
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OFFSET
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3,1
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COMMENTS
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a(n) = 0 iff A000040(n) is a Wolstenholme prime (given by A088164).
For n>2 and p=A000040(n), H(p^2-p) == H(p^2-1) == a(n)*p (mod p^2).
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LINKS
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Table of n, a(n) for n=3..67.
David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Jianqiang Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, Journal of Number Theory, Volume 123, Issue 1, March 2007, Pages 18-26.
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FORMULA
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a(n) = H(p-1)/p^2 mod p = A001008(p-1)/A002805(p-1)/p^2 mod p = A034602(n)/2 mod p = (binomial(2*p-1,p)-1)/(2*p^3) mod p, where p = A000040(n).
a(n) = (-1/3)*B(p-3) mod p, with p=prime(n) and B(n) is the n-th Bernoulli number. - Michel Marcus, Feb 05 2016
a(n) = A087754(n)/4 mod A000040(n).
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PROG
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(PARI) { a(n) = my(p); p=prime(n); ((binomial(2*p-1, p)-1)/2/p^3)%p }
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CROSSREFS
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Cf. A034602, A072984, A087754, A092101, A092103, A092193, A128673.
Sequence in context: A072910 A130482 A239318 * A228945 A348467 A178746
Adjacent sequences: A177780 A177781 A177782 * A177784 A177785 A177786
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KEYWORD
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nonn
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AUTHOR
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Max Alekseyev, May 13 2010
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EXTENSIONS
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Edited by Max Alekseyev, May 16 2010
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STATUS
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approved
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