cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Original entry on oeis.org

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

Views

Author

Alexander Adamchuk, Oct 18 2009

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A000330, A001044, A125294, A125314, A060462, A166604, A166605, A166606, A166607, A166608, A166609, A166610, A334735.
Cf. A067656. - R. J. Mathar, Oct 23 2009

Programs

  • Maple
    q:= k-> is(irem(k!^2, k*(k+1)*(2*k+1)/6)=0):
select(q, [$1..200])[];  # Alois P. Heinz, May 09 2020
  • Mathematica
    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 *)
  • PARI
    isok(k) = ((k!)^2 % (k*(k+1)*(2*k+1)/6)) == 0; \\ Michel Marcus, May 09 2020

Extensions

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

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

Views

Author

Alexander Adamchuk, Oct 18 2009

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Alexander Adamchuk, Oct 18 2009

Keywords

Examples

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

Links

Crossrefs

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

Extensions

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

Views

Author

Alexander Adamchuk, Oct 18 2009

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Alexander Adamchuk, Oct 18 2009

Keywords

Examples

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

Crossrefs

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

Programs

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

Extensions

More terms from Max Alekseyev, Sep 30 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

Views

Author

Alexander Adamchuk, Oct 18 2010

Keywords

Comments

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

Examples

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

Links

Crossrefs

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).

Programs

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

A334734 Denominator of Sum_{k=1..n} k^4 / Product_{k=1..n} k^4.

Original entry on oeis.org

1, 16, 648, 55296, 207360000, 10749542400, 23044331520000, 220242357780480000, 5780040437590917120000, 3538800267912806400000000, 115398507336502660300800000000, 5264387585885382160794255360000000, 1835850718442886445597615718400000000
Offset: 1

Views

Author

Petros Hadjicostas, May 09 2020

Keywords

Examples

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

Crossrefs

Cf. A166604, A181426 (numerators).

Programs

  • PARI
    a(n) = denominator(sum(k=1, n, k^4)/prod(k=1, n, k^4)); \\ Michel Marcus, May 09 2020
Showing 1-7 of 7 results.

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