A238201 Primes p such that numerator(H(floor(p/6))) == 0 (mod p), where H(k) is the k-th harmonic number.

2, 3, 5, 61, 1680023, 7308036881

Offset: 1

Author

T. D. Noe, Feb 24 2014

Keywords

• nonn
• more

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.

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

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

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

Links

• 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 [math.NT], 2014-2015.
• John Blythe Dobson, Calculations relating to some special Harmonic numbers, arXiv:1501.05075 [math.NT], 2015.
• Richard Fischer, Fermatquotient B^(P-1) == 1 (mod P^2).
• 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.

Crossrefs

Cf. A001008, A318761.

Programs

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 *)

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