A126196 -id:A126196 - cp's OEIS frontend

A128671 Least number k > 0 such that k^p does not divide the denominator of generalized harmonic number H(k,p) nor the denominator of alternating generalized harmonic number H'(k,p), where p = prime(n).

20, 94556602, 444, 104, 77, 3504, 1107, 104, 2948, 903, 77, 1752, 77, 104, 370

Offset: 1

Alexander Adamchuk, Mar 24 2007, Mar 26 2007

Generalized harmonic numbers are defined as H(m,k) = Sum_{i=1..m} 1/i^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{i=1..m} (-1)^(i+1)*1/i^k.
a(18)..a(24) = {77,104,77,136,104,370,136}. a(26)..a(27) = {77,104}.
a(n) is currently unknown for n = {16,17,25,...}. See more details in Comments at A128672 and A125581.

			a(2) = A128673(1) = 94556602.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A128670, A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A125581, A126196, A126197, A128672, A128673, A128674, A128675, A128676.

a(n) = A128670(prime(n)).

a(9) = 2948 and a(12) = 1752 from Max Alekseyev

Edited by Max Alekseyev, Feb 20 2019

A126563 Numbers k such that the ratio of A117731(k) and A082687(k) is composite.

Original entry on oeis.org

119, 735, 5145, 36015, 252105, 1764735, 12353145

Offset: 1

Alexander Adamchuk, Mar 12 2007, Jun 09 2007

a(1) = 7*17, a(2) = 3*5*7^2, a(3) = 3*5*7^3.
Corresponding composite terms in A125741 are {119, 49, 49, 49, 49, 49, 49, ...}.
A125741(n) is composite for n = {7, 16, 36, 91, 226, 510, 1131, ...}.

Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Cf. A082687, A117731, A125581, A125740, A125741, A126196, A126197.

h=0; Do[ h=h+1/(n+1)/(2n+1); f=Numerator[n*h]; g=Numerator[h]; If[ !Equal[f, g] && !PrimeQ[f/g], Print[ {n, f/g, FactorInteger[n], FactorInteger[f/g]} ] ], {n, 1, 10000} ]

f(n) = sum(k=1, n, 1/(n+k));
isok(k) = my(fk = f(k), q = numerator(k*fk)/numerator(fk)); (q!=1) && !isprime(q); \\ Michel Marcus, Mar 08 2023

Edited by Max Alekseyev, Jul 12 2019

a(5)-a(7) from Jinyuan Wang, Jul 10 2025

cp's OEIS Frontend

A128671 Least number k > 0 such that k^p does not divide the denominator of generalized harmonic number H(k,p) nor the denominator of alternating generalized harmonic number H'(k,p), where p = prime(n).

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