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 1-8 of 8 results.

A341064 Number of ways to write n as an ordered sum of 4 squarefree numbers.

Original entry on oeis.org

1, 4, 10, 16, 23, 32, 50, 68, 83, 92, 116, 148, 178, 192, 224, 276, 335, 360, 400, 460, 547, 580, 634, 704, 821, 868, 938, 1024, 1162, 1212, 1288, 1392, 1572, 1628, 1742, 1876, 2123, 2172, 2308, 2460, 2761, 2820, 2964, 3176, 3550, 3628, 3778, 4028, 4481, 4528, 4686, 4932, 5513, 5564
Offset: 4

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008683, A008966, A098235, A280210, A308767, A341065, A341066, A341067, A341068, A341069, A341070.

Programs

  • Maple
    b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..57);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    nmax = 57; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

Formula

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^4.

A341066 Number of ways to write n as an ordered sum of 6 squarefree numbers.

Original entry on oeis.org

1, 6, 21, 50, 96, 162, 267, 426, 645, 902, 1218, 1632, 2187, 2826, 3543, 4402, 5547, 6906, 8397, 10032, 12108, 14578, 17298, 20112, 23517, 27534, 32034, 36592, 41892, 48018, 54886, 61758, 69549, 78408, 88365, 98274, 109478, 122058, 136230, 150114, 165759, 183114, 202630, 221484
Offset: 6

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Examples

			G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^6.

Crossrefs

Cf. A005117, A008683, A008966, A098235, A280210, A308902, A341064, A341065, A341067, A341068, A341069, A341070.

Programs

  • Maple
    b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..49);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    nmax = 49; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

A341067 Number of ways to write n as an ordered sum of 7 squarefree numbers.

Original entry on oeis.org

1, 7, 28, 77, 168, 315, 553, 932, 1505, 2282, 3297, 4634, 6447, 8771, 11607, 15029, 19390, 24885, 31500, 39137, 48335, 59584, 73003, 88109, 105525, 126112, 150472, 177632, 208160, 243194, 284102, 329357, 379379, 435477, 500108, 571124, 648998, 735112, 833483, 940765, 1057679
Offset: 7

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Links

Crossrefs

Cf. A005117, A008683, A008966, A098235, A280210, A308952, A341064, A341065, A341066, A341068, A341069, A341070.

Programs

  • Maple
    b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..47);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    nmax = 47; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

Formula

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^7.

A341068 Number of ways to write n as an ordered sum of 8 squarefree numbers.

Original entry on oeis.org

1, 8, 36, 112, 274, 568, 1072, 1912, 3263, 5280, 8128, 12048, 17474, 24824, 34428, 46600, 62163, 82160, 107452, 138392, 176116, 222560, 279756, 348168, 428954, 524848, 639976, 775448, 932376, 1113808, 1326748, 1573656, 1855767, 2175728, 2544048, 2965280, 3441568, 3974744, 4580060
Offset: 8

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008683, A008966, A098235, A280210, A326443, A341064, A341065, A341066, A341067, A341069, A341070.

Programs

  • Maple
    b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..46);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    nmax = 46; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

Formula

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^8.

A341069 Number of ways to write n as an ordered sum of 9 squarefree numbers.

Original entry on oeis.org

1, 9, 45, 156, 423, 963, 1959, 3708, 6669, 11410, 18594, 29052, 44046, 65196, 94284, 133104, 184248, 251406, 338995, 450936, 591885, 768657, 990567, 1265832, 1602273, 2010294, 2506572, 3107136, 3825675, 4676643, 5686347, 6882912, 8290431, 9928305, 11834289, 14052816, 16624846
Offset: 9

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008683, A008966, A098235, A280210, A326522, A341064, A341065, A341066, A341067, A341068, A341070.

Programs

  • Maple
    b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..45);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    nmax = 45; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

Formula

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^9.

A341070 Number of ways to write n as an ordered sum of 10 squarefree numbers.

Original entry on oeis.org

1, 10, 55, 210, 625, 1552, 3400, 6840, 12960, 23330, 40028, 65740, 104230, 160670, 241640, 354772, 509620, 718980, 999645, 1370720, 1853903, 2476250, 3274110, 4289810, 5568820, 7162184, 9138045, 11579180, 14574755, 18215900, 22619016, 27929990, 34311845, 41921710, 50946945
Offset: 10

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008683, A008966, A098235, A280210, A326626, A341064, A341065, A341066, A341067, A341068, A341069.

Programs

  • Maple
    b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..44);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    nmax = 44; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^10, {x, 0, nmax}], x] // Drop[#, 10] &

Formula

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^10.

A341074 Number of partitions of n into 5 distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 0, 2, 3, 3, 3, 5, 8, 9, 8, 11, 15, 16, 16, 22, 27, 30, 31, 38, 46, 48, 49, 57, 72, 73, 76, 90, 107, 109, 112, 128, 151, 156, 160, 182, 214, 220, 224, 250, 290, 297, 306, 335, 387, 399, 409, 442, 503, 517, 529, 572, 641, 660, 676, 726, 809, 829, 846, 903
Offset: 17

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008966, A098236, A307835, A308840, A341065, A341073, A341075, A341095, A341096, A341097, A341098.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(numtheory[issqrfree](i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 5):
seq(a(n), n=17..77);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[SquareFreeQ[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 5];
Table[a[n], {n, 17, 77}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n,{5}],?(Length[Union[#]]==5&&AllTrue[#,SquareFreeQ]&)],{n,17,80}] (* _Harvey P. Dale, Sep 05 2023 *)

A347780 Number of compositions (ordered partitions) of n into at most 5 squarefree parts.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 45, 71, 105, 151, 214, 291, 379, 473, 593, 744, 919, 1095, 1301, 1563, 1884, 2203, 2536, 2929, 3427, 3929, 4433, 4979, 5692, 6422, 7158, 7904, 8863, 9844, 10830, 11810, 13078, 14378, 15706, 17007, 18718, 20424, 22165, 23803, 26025
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2021

Keywords

Crossrefs

Cf. A005117, A280194, A341065, A347777, A347778, A347779.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,5,Select[Range@n,SquareFreeQ]],1],{n,0,45}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)
Showing 1-8 of 8 results.

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