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A238201 Primes p such that numerator(H(floor(p/6))) == 0 (mod p), where H(k) is the k-th harmonic number. 0
2, 3, 5, 61, 1680023, 7308036881 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The H function is 0 for the first three primes. The term 61 comes from Schwindt's paper. The terms 1680023 and 7308036881 come from Dobson's paper.

LINKS

Table of n, a(n) for n=1..6.

Karl Dilcher and Ladislav Skula, A new criterion for the first case of Fermat’s Last Theorem, Math. Comp. 64 (1995) 363-392.

John Blythe Dobson, Extended calculations of a special harmonic number, arxiv 1402.5680, 2014.

John Blythe Dobson, Calculations relating to some special Harmonic numbers, arXiv:1501.05075 [math.NT], 2015.

Emma Lehmer, On Congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. 39 (1938) 350-360.

H. Schwindt, Three summation criteria for Fermat’s Last Theorem, Math. Comp. 40 (1983) 715-716.

FORMULA

Let q_2 = (2^(p-1) - 1)/p and q_3 = (3^(p-1) - 1)/p. Then, as proved by Emma Lehmer, H(floor(p/6)) == -2*q_2 - (3/2)*q_3 (mod p) when p > 3. This congruence provides an efficient means of detecting when H(floor(p/6)) vanishes mod p. - John Blythe Dobson, Mar 01 2014

MATHEMATICA

Select[Prime[Range[1000]], Mod[Numerator[HarmonicNumber[Floor[#/6]]], #] == 0 &]

Select[Prime[Range[1000]], Divisible[Numerator[HarmonicNumber[Quotient[#, 6]]], #] &] (* Jan Mangaldan, May 07 2014 *)

PROG

(PARI) is(n)=my(H=sum(i=1, n\6, 1/i)); numerator(H)%n==0 && isprime(n) \\ Charles R Greathouse IV, Mar 02 2014

CROSSREFS

Cf. A001008.

Sequence in context: A029961 A083665 A214752 * A084839 A259382 A103110

Adjacent sequences:  A238198 A238199 A238200 * A238202 A238203 A238204

KEYWORD

nonn,more

AUTHOR

T. D. Noe, Feb 24 2014

STATUS

approved

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Last modified October 23 18:50 EDT 2018. Contains 316530 sequences. (Running on oeis4.)