A322434 -id:A322434 - cp's OEIS frontend

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

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

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

			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.

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

Cf. A076638, A001008.

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

More terms from Amiram Eldar, Dec 04 2018

A326227 Indices of nonsquarefree numerators (A001008) of harmonic numbers H(n) = Sum_{k=1..n} 1/k.

Original entry on oeis.org

4, 6, 7, 10, 12, 16, 18, 22, 28, 29, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276

Offset: 1

M. F. Hasler, Jul 03 2019

It appears that the first term of A001008 having a cubic factor is A001008(848) = 11^3 * 1871 * C359.

By Wolstenholme's Theorem, p^2 divides A001008(p-1) whenever p >= 5 is prime (cf. A076637); see A308968 for illustration. Therefore, A006093 \ {1, 2} (primes - 1) is a subsequence. (Thanks to Bernard Schott.)

Cf. A001008, A076637, A322434.

Cf. A308967 (number of prime factors), A308968 (table of factorization), A308969 (table of prime divisors), A308970 & A308971 (smallest & largest prime factor) of A001008(n).

is_A326227(n)={n>3&&vecmax(factor(A001008(n))[,2])>1} \\ Add ,0 in factor() for much faster but possibly incorrect results [false negative].
for(n=1,oo, is_A326227(n) && print1(n","))

cp's OEIS Frontend

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

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