A125294 -id:A125294 - cp's OEIS frontend

A166602 Numbers k such that Sum_{i=1..k} i^2 divides Product_{i=1..k} i^2.

1, 7, 13, 17, 19, 24, 25, 27, 31, 32, 34, 37, 38, 43, 45, 47, 49, 55, 57, 59, 61, 62, 64, 67, 71, 73, 76, 77, 79, 80, 84, 85, 87, 91, 92, 93, 94, 97, 101, 103, 104, 107, 109, 110, 115, 117, 118, 121, 122, 123, 124, 127, 129, 132, 133, 137, 139, 142, 143, 144, 145, 147

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

Product_{i=1..k} i^2 = (k!)^2 and Sum_{i=1..k} i^2 = k*(k+1)*(2*k+1)/6. - J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

			a(2) = A125314(2) = 7.

J. Mulder, Table of n, a(n) for a(n) below 20000

Cf. A000330, A001044, A125294, A125314, A060462, A166604, A166605, A166606, A166607, A166608, A166609, A166610, A334735.

Cf. A067656. - R. J. Mathar, Oct 23 2009

q:= k-> is(irem(k!^2, k*(k+1)*(2*k+1)/6)=0):
select(q, [$1..200])[];  # Alois P. Heinz, May 09 2020

Cases[Range[2, 5000], k_ /; Divisible[Factorial[k - 1]^2, 1/6 (-1 + k) k (-1 + 2 k)]] - 1 (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

isok(k) = ((k!)^2 % (k*(k+1)*(2*k+1)/6)) == 0; \\ Michel Marcus, May 09 2020

Terms below 5000 by J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

More terms copied from the b-file by R. J. Mathar, Feb 14 2010

A181426 Numerator of Sum_{k=1..n} k^4 / Product_{k=1..n} k^4.

Original entry on oeis.org

1, 17, 49, 59, 979, 91, 167, 731, 5111, 517, 1817, 6071, 109, 18241, 22289, 2771, 131, 28823, 67, 51619, 11911, 34891, 15557, 257, 1949, 22313, 2267, 14123, 153931, 5273999, 1, 3167, 45091, 3569, 268309, 126947, 4217, 127, 369641, 201679, 85739

Offset: 1

Alexander Adamchuk, Oct 18 2010

a(n) = 1 for n = {1, 31, 59, 94, 104, 122, 133, 181, 206, 223, ...} = A166604.

			The first few fractions are 1, 17/16, 49/648, 59/55296, 979/207360000, 91/10749542400, 167/23044331520000, ... = A181426/A334734.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A090585 = Numerator of Sum/Product of first n numbers.
Cf. A125294 = Numerator of Sum/Product of squares of first n numbers.
Cf. A166604, A334734 (denominators).

Table[Numerator[Sum[ k^4, {k, 1, n}] / Product[k^4, {k, 1, n}]], {n, 1, 100}]
With[{c=Range[50]^4},Numerator[Accumulate[c]/FoldList[Times,c]]] (* Harvey P. Dale, Jul 03 2025 *)

a(n) = numerator(sum(k=1, n, k^4)/prod(k=1, n, k^4)); \\ Michel Marcus, May 09 2020

A334735 Denominator of Sum_{k=1..n} k^2 / Product_{k=1..n} k^2.

Original entry on oeis.org

1, 4, 18, 96, 2880, 518400, 181440, 135475200, 8778792960, 376233984000, 72425041920000, 4588850656051200, 47345284546560000, 217144413044342784000, 42750306318104985600000, 4974581098834034688000000, 70875936417673484697600000, 13663463022599094380003328000000

Offset: 1

Petros Hadjicostas, May 09 2020

			The first few fractions are 1, 5/4, 7/18, 5/96, 11/2880, 91/518400, 1/181440, 17/135475200, 19/8778792960, ... = A125294/A334735.

Cf. A125294 (numerators), A166602.

Table[Denominator[n*(n + 1)*(2*n + 1)/(6*(n!)^2)], {n, 1, 18}] (* Amiram Eldar, May 09 2020 *)

a(n) = denominator(sum(k=1, n, k^2)/prod(k=1, n, k^2)); \\ Michel Marcus, May 09 2020

a(n) = denominator(n*(n + 1)*(2*n + 1)/6/(n!)^2).

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A166602 Numbers k such that Sum_{i=1..k} i^2 divides Product_{i=1..k} i^2.

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A181426 Numerator of Sum_{k=1..n} k^4 / Product_{k=1..n} k^4.

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A334735 Denominator of Sum_{k=1..n} k^2 / Product_{k=1..n} k^2.

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