A166602 -id:A166602 - cp's OEIS frontend

A166604 Numbers k such that Sum_{i=1..k} i^4 divides Product_{i=1..k} i^4.

1, 31, 59, 94, 104, 122, 133, 181, 206, 223, 244, 248, 283, 298, 318, 342, 356, 401, 406, 421, 422, 439, 444, 449, 451, 469, 479, 493, 496, 507, 528, 532, 536, 541, 555, 597, 631, 637, 643, 668, 701, 706, 712, 717, 721, 722, 754, 762, 795, 797, 801, 815, 842

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(4) = 31.

Harvey P. Dale, Table of n, a(n) for n = 1..3000

Cf. A125314, A166602, A060462, A166605, A166606, A166607, A166608, A166609, A166610, A181426, A334734.

k = s = 1; p = 1; lst = {}; While[k < 1000, If[ Mod[p, s] == 0, AppendTo[lst, k]]; k++; s = s + k^4; p = p*k^4]; lst (* Robert G. Wilson v, Nov 02 2009 *)
Module[{nn=1000,c},c=Range[nn]^4;Select[Range[nn],Divisible[Times@@ Take[ c,#], Total[Take[c,#]]]&]] (* Harvey P. Dale, Dec 18 2013 *)

a(15)-a(53) from Robert G. Wilson v, Nov 02 2009

A166605 Numbers k such that Sum_{i=1..k} i^5 divides Product_{i=1..k} i^5.

Original entry on oeis.org

1, 13, 64, 95, 111, 118, 123, 133, 134, 140, 151, 177, 199, 217, 229, 242, 255, 264, 274, 281, 302, 305, 325, 333, 338, 354, 359, 365, 376, 394, 411, 414, 431, 433, 440, 472, 475, 477, 489, 514, 525, 528, 529, 537, 547, 569, 574, 583, 584, 585, 589, 594, 615

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(5) = 13.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A125314, A166602, A060462, A166604, A166606, A166607, A166608, A166609, A166610.

k = s = 1; p = 1; lst = {}; While[k < 100000, If[Mod[p, s] == 0, AppendTo[lst, k]]; k++; s = s + k^5; p = p*k^5]; lst  (* G. C. Greubel, May 18 2016 *)
Module[{nn=1000,i5},i5=Range[nn]^5;Position[Table[Times@@Take[i5,n]/Total[Take[i5,n]],{n,nn}],?IntegerQ]]//Flatten (* _Harvey P. Dale, Jan 18 2025 *)

More terms from Max Alekseyev, Sep 29 2010

A166609 Numbers k such that Sum_{i=1..k} i^9 divides Product_{i=1..k} i^9.

Original entry on oeis.org

1, 1441, 1715, 11706, 16741, 18435, 23793, 29927, 32071, 33932, 45768, 45831, 47103, 47215, 48257, 55743, 56007, 61976, 62773, 64841, 68561, 70853, 70880, 81624, 83526, 86243, 87529, 88162, 91054, 91395, 92288, 92933, 94211, 98982

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(9) = 1441.

David A. Corneth, Table of n, a(n) for n = 1..3003

Cf. A125314, A166602, A060462, A166604, A166605, A166606, A166607, A166608, A166610.

More terms from Max Alekseyev, Sep 30 2010

A166606 Numbers k such that Sum_{i=1..k} i^6 divides Product_{i=1..k} i^6.

Original entry on oeis.org

1, 1556, 1640, 3907, 5642, 6205, 7238, 8311, 10350, 11551, 12499, 13371, 13812, 17524, 17589, 18162, 18790, 21569, 21573, 22381, 22544, 23809, 24312, 24416, 24598, 24629, 25247, 25463, 26093, 26583, 26829, 27091, 27098, 28646, 28804

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(6) = 1556.

Cf. A125314, A166602, A060462, A166604, A166605, A166607, A166608, A166609, A166610.

k = s = 1; p = 1; lst = {}; While[k < 5000, If[Mod[p, s] == 0, AppendTo[lst, k]]; k++; s = s + k^6; p = p*k^6]; lst (* G. C. Greubel, May 18 2016 *)

More terms from Max Alekseyev, Sep 30 2010

A166607 Numbers k such that Sum_{i=1..k} i^7 divides Product_{i=1..k} i^7.

Original entry on oeis.org

1, 733, 1637, 2096, 2367, 4231, 5674, 5839, 7585, 8344, 13719, 13753, 14983, 15151, 15197, 15257, 15757, 16595, 17305, 18791, 20701, 21442, 23652, 23738, 24519, 24789, 25474, 25916, 25933, 27474, 27487, 29185, 31455, 32846, 32950, 33421

Offset: 1

Alexander Adamchuk, Oct 18 2009

nonn

			a(2) = A125314(7) = 733.

Cf. A123314, A166602, A060462, A166604, A166605, A166606, A166608, A166609, A166610.

With[{c=Range[40000]^7},Flatten[Position[#[[1]]/#[[2]]&/@Thread[ {Rest[ FoldList[ Times,1,c]],Accumulate[c]}],?IntegerQ]]] (* _Harvey P. Dale, Nov 16 2014 *)

More terms from Max Alekseyev, Sep 30 2010

A125294 Numerator of (Sum_{k=1..n} k^2) / (Product_{k=1..n} k^2).

Original entry on oeis.org

1, 5, 7, 5, 11, 91, 1, 17, 19, 11, 23, 13, 1, 29, 31, 17, 1, 703, 1, 41, 43, 23, 47, 1, 1, 53, 1, 29, 59, 1891, 1, 1, 67, 1, 71, 2701, 1, 1, 79, 41, 83, 43, 1, 89, 1, 47, 1, 97, 1, 101, 103, 53, 107, 109, 1, 113, 1, 59, 1, 61, 1, 1, 127, 1, 131, 67, 1, 137, 139, 71, 1, 73, 1, 149

Offset: 1

Alexander Adamchuk, Jan 17 2007

All a(n) are either 1, semiprime or prime.
a(n) = 1 for n = 1 and n = {7, 13, 17, 19, 24, 25, 27, 31, 32, 34, 37, 38, 43, 45, 47, 49, ...} = A067656 = numbers n such that n!*B(2*n) is an integer, where the B(2*n)'s are the Bernoulli numbers.
p divides a(p-1) for prime p > 3. p divides a((p-1)/2) for prime p > 3.
a(p-1) = p*(2p-1) is a semiprime hexagonal number for prime p = {7, 19, 31, 37, 79, 97, 139, 157, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, ...} = A005382(n) for n > 2, where A005382(n) are the numbers n such that n and 2*n-1 are primes.
a(p-1) = p for prime p = {5, 11, 13, 17, 23, 29, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, ...} = primes that do not belong to A005382(n).
a((p-1)/2) = p for prime p = {5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 259, 271, 281, 283, 293, 307, 311, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 401, ...}, which is apparently the union of {5} and A034849(n).

			The first few fractions are 1, 5/4, 7/18, 5/96, 11/2880, 91/518400, 1/181440, 17/135475200, 19/8778792960, ... = A125294/A334735. - _Petros Hadjicostas_, May 09 2020

Cf. A005382, A034849, A067656, A166602, A334735 (denominators).

Table[Numerator[n(n+1)(2n+1)/6/(n!)^2],{n,1,500}]

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

a(n) = numerator((Sum_{k=1..n} k^2) / (Product_{k=1..n} k^2)).

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

A067656 Numbers n such that n!*B(2n) is an integer, where B(2n) are the Bernoulli numbers.

Original entry on oeis.org

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

Benoit Cloitre, Feb 03 2002

nonn

A045979(n), Bernoulli numbers with denominators 6, are included in the sequence.

Also numbers n such that both n+1 and 2n+1 are not prime. - Alexander Adamchuk, Oct 05 2006

Alexander Adamchuk, Oct 05 2006, Table of n, a(n) for n = 1..565

Cf. A000330, A001044.

Cf. A166602. - R. J. Mathar, Feb 14 2010

Select[Range[2,1000],Numerator[ #(#+1)(2#+1)/6/#!^2]==1&] (* Alexander Adamchuk, Oct 05 2006 *)
Select[Range[1000],!PrimeQ[ #+1]&&!PrimeQ[2#+1]&] (* Alexander Adamchuk, Oct 05 2006 *)

Also numbers n>1 such that A000330[n] = Sum[k^2,{k,1,n}] = n(n+1)(2n+1)/6 divides A001044[n] = Product[k^2,{k,1,n}] = (n!)^2. Also numbers n>1 such that Numerator[n(n+1)(2n+1)/6 /(n!)^2] = 1. - Alexander Adamchuk, Oct 05 2006

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).

cp's OEIS Frontend

A166604 Numbers k such that Sum_{i=1..k} i^4 divides Product_{i=1..k} i^4.

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A166605 Numbers k such that Sum_{i=1..k} i^5 divides Product_{i=1..k} i^5.

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A166609 Numbers k such that Sum_{i=1..k} i^9 divides Product_{i=1..k} i^9.

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A166606 Numbers k such that Sum_{i=1..k} i^6 divides Product_{i=1..k} i^6.

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A166607 Numbers k such that Sum_{i=1..k} i^7 divides Product_{i=1..k} i^7.

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A125294 Numerator of (Sum_{k=1..n} k^2) / (Product_{k=1..n} k^2).

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A067656 Numbers n such that n!*B(2n) is an integer, where B(2n) are the Bernoulli numbers.

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

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