A177734 Largest k such that prime(n) divides the numerator of the k-th harmonic number (=A001008(k)).

22, 24, 102728, 1011849771855214912968404217247, 168, 288, 848874360, 528, 695552, 886725671, 50641, 1680, 2359785, 10776888210, 414839198, 42176361744, 226972, 4488, 9094138358932, 5328, 6240

Offset: 2

Author

Max Alekseyev, May 12 2010

Keywords

• hard
• nonn

Comments

For p = prime(n), Boyd defines J_p to be the set of numbers k such that p divides A001008(k). This sequence gives the largest element of J_p. The smallest element of J_p is given by A072984. The size of J_p is given by A092103.

Term a(23) is too large to include, see b-file. - Max Alekseyev, Apr 04 2025

Links

• Max Alekseyev, Table of n, a(n) for n = 2..220
• David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302. [WARNING: Table 2 contains miscalculations for p=19, 47, 59, ... - _Max Alekseyev_, Feb 10 2016]
• Leonardo Carofiglio, Giacomo Cherubini, and Alessandro Gambini, On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers, arXiv:2503.15714 [math.NT], 2025.

Crossrefs

Cf. A072984, A092103, A092193.

Formula

For p = prime(n) in A092101, a(n) = p^2 - 1.

Extensions

a(5) computed by Boyd.

a(8)-a(22) from Max Alekseyev, Oct 23 2012