A119682 -id:A119682 - cp's OEIS frontend

A357555 a(n) is the numerator of Sum_{d|n} (-1)^(d+1) / d^2.

1, 3, 10, 11, 26, 5, 50, 43, 91, 39, 122, 55, 170, 75, 52, 171, 290, 91, 362, 143, 500, 183, 530, 215, 651, 255, 820, 275, 842, 13, 962, 683, 1220, 435, 52, 1001, 1370, 543, 1700, 559, 1682, 125, 1850, 61, 2366, 795, 2210, 95, 2451, 1953, 2900, 935, 2810, 205, 3172

Offset: 1

Ilya Gutkovskiy, Oct 03 2022

			1, 3/4, 10/9, 11/16, 26/25, 5/6, 50/49, 43/64, 91/81, 39/50, 122/121, ...

Cf. A017667, A064027, A098987, A119682, A321543, A357556 (denominators).

Table[Sum[(-1)^(d + 1)/d^2, {d, Divisors[n]}], {n, 1, 55}] // Numerator
nmax = 55; CoefficientList[Series[Sum[(-1)^(k + 1) x^k/(k^2 (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest

a(n) = numerator(sumdiv(n, d, (-1)^(d+1)/d^2)); \\ Michel Marcus, Oct 03 2022

from sympy import divisors
from fractions import Fraction
def a(n): return sum(Fraction((-1)**(d+1), d*d) for d in divisors(n, generator=True)).numerator
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 03 2022

Numerators of coefficients in expansion of Sum_{k>=1} (-1)^(k+1) * x^k / (k^2 * (1 - x^k)).

A119784 Numerator of Sum[Sum[(-1)^(i+1)1/(ij)^2, {i, 1, n}], {j, 1, n}].

Original entry on oeis.org

1, 15, 1519, 23575, 15907111, 5224037, 13034567237, 206747823917, 51412399560631, 10224262569323, 152012226379719803, 151406414338463843, 4372018721271579430163, 4359099317599237566323, 21975413631947472215359

Offset: 1

Alexander Adamchuk, Jun 25 2006

Prime p divides a((p-1)/2) for p>3. p^2 divides a(p-1) for prime p>3.

Cf. A007406, A119682.

Numerator[Table[Sum[Sum[(-1)^(i+1)*1/(i*j)^2, {i, 1, n}], {j, 1, n}],{n,1,20}]]

a(n) = numerator[Sum[Sum[(-1)^(i+1)*1/(i*j)^2, {i, 1, n}], {j, 1, n}]]. a(n) = A119682(n)*A007406(n).

A125504 Smallest number k such that the numerator of alternating generalized harmonic number H'(k,n) = Sum[ (-1)^(i+1) * 1/i^n, {i,1,k} ] is a prime.

Original entry on oeis.org

3, 2, 2, 3, 2, 19, 2, 146, 87, 3, 16, 3, 2, 249, 15, 87, 2, 699, 2

Offset: 1

Alexander Adamchuk, Dec 28 2006, Jan 31 2007

a(n) = 2 for n in A000043.
a(n) = 3 for n in {1,4,10,12,24,27,39,...}.
a(n) = 5 for n in {26,76,132,205,238,...}.
a(n) = 9 for n in {100,200,...}.
a(n) = 15 for n in {15,33,65,...}.
a(21) = 18. a(22) = 13. a(41) = 6. a(72) = 11. a(173) = 8.
a(20) > 2100 - Max Alekseyev, Jul 07 2009

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A058313, A119682, A120296, A001008, A000043.

More terms from Max Alekseyev, Jul 07 2009

A128820 Numerator of alternating generalized harmonic number H'(p-1,2p) = Sum_{k=1..p-1} (-1)^(k+1)/k^(2*p) divided by p^2 for prime p > 2.

Original entry on oeis.org

7, 2474315503, 53305712401979540402437, 5597916593064896381208777124641713285719656398067086247546781015747740847, 192635872080422175485338764164035657976855166649911323825254242037669356649787653784405726270977624462974729613783

Offset: 2

Alexander Adamchuk, Apr 10 2007

nonn

Alternating generalized harmonic number is H'(n,m) = Sum_{k=1..n} (-1)^(k+1)*1/k^m. Numerator of H'(p-1,2n) is divisible by p for all integers n > 0 and primes p > 2. Numerator of H'(p-1,2p) is divisible by p^2 for prime p > 2.

			prime(2) = 3; a(2) = numerator(1 - 1/2^6) / 3^2 = 63/9 = 7.
prime(3) = 5; a(3) = numerator(1 - 1/2^10 + 1/3^10 - 1/4^10) / 5^2 = 61857887575/25 = 2474315503.

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

Cf. A119722 (numerator of generalized harmonic number H(p-1, p) = Sum_{k=1..p-1} 1/k^p divided by p^3 for prime p>3).

Cf. A001008, A119682, A120296.

Table[ Numerator[ Sum[(-1)^(k+1)*1/k^(2*Prime[n]), {k,1,Prime[n]-1} ] ] / Prime[n]^2, {n,2,10} ]

a(n) = numerator(Sum_{k=1..prime(n)-1} (-1)^(k+1)/k^(2*prime(n))) / prime(n)^2 for n > 1.

cp's OEIS Frontend

A357555 a(n) is the numerator of Sum_{d|n} (-1)^(d+1) / d^2.

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A119784 Numerator of Sum[Sum[(-1)^(i+1)1/(ij)^2, {i, 1, n}], {j, 1, n}].

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A125504 Smallest number k such that the numerator of alternating generalized harmonic number H'(k,n) = Sum[ (-1)^(i+1) * 1/i^n, {i,1,k} ] is a prime.

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A128820 Numerator of alternating generalized harmonic number H'(p-1,2p) = Sum_{k=1..p-1} (-1)^(k+1)/k^(2*p) divided by p^2 for prime p > 2.

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cp's OEIS Frontend

A357555 a(n) is the numerator of Sum_{d|n} (-1)^(d+1) / d^2.

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Author

Keywords

Examples

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Programs

Mathematica

PARI

Python

Formula

A119784 Numerator of Sum[Sum[(-1)^(i+1)*1/(i*j)^2, {i, 1, n}], {j, 1, n}].

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Formula

A125504 Smallest number k such that the numerator of alternating generalized harmonic number H'(k,n) = Sum[ (-1)^(i+1) * 1/i^n, {i,1,k} ] is a prime.

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Author

Keywords

Comments

Links

Crossrefs

Extensions

A128820 Numerator of alternating generalized harmonic number H'(p-1,2p) = Sum_{k=1..p-1} (-1)^(k+1)/k^(2*p) divided by p^2 for prime p > 2.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

Formula

A119784 Numerator of Sum[Sum[(-1)^(i+1)1/(ij)^2, {i, 1, n}], {j, 1, n}].