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%I #42 Jan 24 2021 14:55:53
%S 2,3,5,61,1680023,7308036881
%N Primes p such that numerator(H(floor(p/6))) == 0 (mod p), where H(k) is the k-th harmonic number.
%C 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.
%C 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
%C Also {2, 3} union {primes p : p^2 divides 2^(p-2) + 3^(p-2) + 6^(p-2) - 1}. Except for the term 3, p is a term of this sequence if and only if p^2 is in A318761. There are no more terms up to 7*10^10. - _Jianing Song_, Dec 26 2018
%C More generally (see Lehmer's paper, p. 352, eq. 13), if p^2 divides 2^(p-2) + 3^(p-2) + 6^(p-2) - 1, then it also divides 2^(k(p-1)-1) + 3^(k(p-1)-1) + 6^(k(p-1)-1) - 1 for any natural number k. Also, since H(floor(p/6)) == -2*q_2 - (3/2)*q_3 == -q_4 - (1/2)q_27 == -(1/2)(q_16 + q_27) == -(1/2)q_432 (mod p), the terms of this sequence greater than 3 coincide with the values of p that divide q_432, and can be found in Richard Fischer's list of vanishing Fermat quotients, which extended to 1.31*10^14 at the last revision of 19 December 2020. - _John Blythe Dobson_, Jan 02 2021
%H Karl Dilcher and Ladislav Skula, <a href="http://dx.doi.org/10.1090/S0025-5718-1995-1248969-6">A new criterion for the first case of Fermat’s Last Theorem</a>, Math. Comp. 64 (1995) 363-392.
%H John Blythe Dobson, <a href="http://arxiv.org/abs/1402.5680">Extended calculations of a special harmonic number</a>, arxiv 1402.5680 [math.NT], 2014-2015.
%H John Blythe Dobson, <a href="http://arxiv.org/abs/1501.05075">Calculations relating to some special Harmonic numbers</a>, arXiv:1501.05075 [math.NT], 2015.
%H Richard Fischer, <a href="http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort.txt">Fermatquotient B^(P-1) == 1 (mod P^2)</a>.
%H Emma Lehmer, <a href="http://www.jstor.org/stable/1968791">On Congruences involving Bernoulli numbers and the quotients of Fermat and Wilson</a>, Ann. of Math. 39 (1938) 350-360.
%H H. Schwindt, <a href="http://dx.doi.org/10.1090/S0025-5718-1983-0689484-X">Three summation criteria for Fermat’s Last Theorem</a>, Math. Comp. 40 (1983) 715-716.
%t Select[Prime[Range[1000]], Mod[Numerator[HarmonicNumber[Floor[#/6]]], #] == 0 &]
%t Select[Prime[Range[1000]], Divisible[Numerator[HarmonicNumber[Quotient[#, 6]]], #] &] (* _Jan Mangaldan_, May 07 2014 *)
%o (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
%Y Cf. A001008, A318761.
%K nonn,more
%O 1,1
%A _T. D. Noe_, Feb 24 2014
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