A076637 Numerators of harmonic numbers when these numerators are divisible by squares of primes >= 5 in the case of Wolstenholme's Theorem.

25, 49, 7381, 86021, 2436559, 14274301, 19093197, 315404588903, 9304682830147, 54801925434709, 2078178381193813, 12309312989335019, 5943339269060627227, 14063600165435720745359, 254381445831833111660789, 15117092380124150817026911

Offset: 1

Author

Michael Gilleland (megilleland(AT)yahoo.com), Oct 23 2002

Keywords

• nonn
• frac

Comments

By Wolstenholme's Theorem, if p prime >= 5, the numerator of the harmonic number H_{p-1} is always divisible by p^2. The obtained quotients are in A061002. - Bernard Schott, Dec 02 2018

The numbers 363, numerator of H_7 and 9227046511387, numerator of H_{29}, which have been found by Amiram Eldar and Michel Marcus, are also divisible by prime squares, respectively by 11^2 and 43^2, but not in the case of Wolstenholme's Theorem. So, a new sequence A322434 is created with all the numerators of Harmonic numbers which are divisible by any prime square >= 5. - Bernard Schott, Dec 08 2018

Examples

			25 is a term because the numerator of the harmonic number H_4 = 1 + 1/2+ 1/3 + 1/4 = 25/12 is divisible by the square of 5;
49 is a term because the numerator of the harmonic number H_6 = 1 + 1/2+ 1/3 + 1/4 + 1/5 + 1/6 = 49/20 is divisible by the square of 7.
		

Links

• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

Crossrefs

Cf. A076638, A001008.

Programs

Mathematica

a[p_] := Numerator[HarmonicNumber[p - 1]]; a /@ Prime@Range[3, 20] (* Amiram Eldar, Dec 08 2018 *)

Extensions

More terms from Amiram Eldar, Dec 04 2018