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 1-10 of 11 results. Next

A098235 Number of ways to write n as a sum of two ordered positive squarefree numbers.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 6, 4, 3, 4, 7, 6, 5, 6, 10, 8, 8, 6, 11, 8, 9, 8, 14, 10, 9, 10, 13, 10, 9, 10, 16, 12, 13, 12, 22, 14, 13, 14, 22, 16, 15, 18, 25, 20, 15, 16, 26, 20, 16, 14, 27, 20, 20, 14, 26, 20, 21, 18, 29, 22, 21, 22, 30, 22, 21, 22, 35, 24, 25, 22, 42, 26, 27, 26, 39
Offset: 1

Views

Author

Ralf Stephan, Aug 31 2004

Keywords

Comments

a(n) ~ n * Prod[p prime, (1-2/p^2) * Prod[p^2|n, (p^2-1)/(p^2-2)]].

Examples

			a(12)=7 because 12=1+11=2+10=5+7=6+6=7+5=10+2=11+1.

Links

Crossrefs

Cf. A005117, A098236, A071068.

Programs

  • Mathematica
    Join[{0}, Table[Sum[(MoebiusMu[k]*MoebiusMu[n - k + 1])^2, {k, 1, n}], {n, 1, 50}]] (* G. C. Greubel, Dec 28 2016 *)
  • PARI
    a(n) = sum(k=1, n-1, (moebius(k)*moebius(n-k))^2) \\ Indranil Ghosh, Mar 10 2017
  • PARI
    a(n)=my(s); forsquarefree(k=1, n-1, s+=issquarefree(n-k)); s \\ Charles R Greathouse IV, Jan 08 2018

Formula

a(n) = Sum_{k=1..n-1} (mu(k)*mu(n-k))^2. - Benoit Cloitre, Sep 24 2006
a(n) = Sum_{k=1..n-1} ( A008966(k)*A008966(n-k) ). - Reinhard Zumkeller, Nov 04 2009
G.f.: ( Sum_{k>=1} mu(k)^2*x^k )^2, where mu(k) is the Moebius function (A008683). - Ilya Gutkovskiy, Dec 28 2016

A307835 Number of partitions of n into 3 distinct squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 2, 2, 3, 5, 4, 4, 5, 9, 8, 8, 9, 12, 11, 11, 12, 16, 15, 15, 17, 21, 19, 18, 20, 25, 24, 22, 28, 33, 32, 28, 33, 40, 37, 35, 40, 50, 47, 42, 48, 58, 56, 48, 56, 65, 66, 57, 63, 73, 73, 65, 70, 82, 80, 74, 81, 92, 90, 80, 92, 102, 102, 88, 104, 116, 116
Offset: 0

Views

Author

Ilya Gutkovskiy, May 01 2019

Keywords

Examples

			a(15) = 4 because we have [11, 3, 1], [10, 3, 2], [7, 6, 2] and [7, 5, 3].

Links

Crossrefs

Cf. A005117, A008683, A087188, A098236, A125688, A307815, A307825.

Programs

  • Mathematica
    Table[Count[IntegerPartitions[n, {3}], _?(And[UnsameQ @@ #, AllTrue[#, SquareFreeQ[#] &]] &)], {n, 0, 75}]

Formula

a(n) = [x^n y^3] Product_{k>=1} (1 + mu(k)^2*y*x^k).

A341073 Number of partitions of n into 4 distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 3, 2, 5, 7, 8, 7, 11, 13, 15, 13, 17, 20, 23, 21, 28, 33, 34, 32, 40, 44, 47, 44, 55, 63, 66, 62, 75, 84, 87, 81, 98, 110, 115, 109, 127, 144, 148, 140, 159, 180, 186, 177, 199, 220, 231, 217, 241, 264, 275, 262, 290, 317, 325, 314, 343, 376, 382, 368, 403
Offset: 11

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Links

Crossrefs

Cf. A005117, A008966, A098236, A307835, A308767, A341064, A341074, 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, 4):
seq(a(n), n=11..75);  # 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, 4];
Table[a[n], {n, 11, 75}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)

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 *)

A341075 Number of partitions of n into 6 distinct squarefree parts.

Original entry on oeis.org

1, 0, 0, 1, 2, 2, 2, 3, 6, 5, 6, 7, 12, 12, 15, 18, 26, 26, 28, 34, 44, 46, 50, 60, 77, 79, 86, 98, 122, 126, 134, 154, 188, 196, 207, 236, 277, 292, 305, 343, 400, 423, 443, 492, 567, 596, 624, 686, 779, 819, 856, 938, 1052, 1108, 1149, 1255, 1394, 1463, 1515, 1646, 1818
Offset: 24

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008966, A098236, A307835, A308902, A341066, A341073, A341074, 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, 6):
seq(a(n), n=24..84);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    Table[Length[Select[IntegerPartitions[n,{6}],Length[Union[#]]==6&&AllTrue[ #,SquareFreeQ]&]],{n,24,90}] (* Harvey P. Dale, Jan 16 2022 *)

A341095 Number of partitions of n into 7 distinct squarefree parts.

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 2, 4, 6, 7, 7, 10, 14, 15, 15, 21, 28, 32, 32, 44, 53, 60, 60, 76, 93, 103, 107, 131, 157, 172, 178, 211, 247, 273, 283, 333, 384, 423, 439, 507, 577, 629, 657, 747, 846, 917, 960, 1078, 1211, 1306, 1362, 1521, 1691, 1822, 1898, 2103, 2322, 2494, 2596, 2850, 3134
Offset: 34

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008966, A098236, A307835, A308952, A341067, A341073, A341074, A341075, 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, 7):
seq(a(n), n=34..94);  # 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, 7];
Table[a[n], {n, 34, 94}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

A341096 Number of partitions of n into 8 distinct squarefree parts.

Original entry on oeis.org

1, 0, 1, 2, 2, 1, 3, 4, 5, 5, 8, 12, 14, 13, 18, 24, 28, 27, 38, 49, 55, 57, 71, 89, 99, 104, 125, 156, 171, 183, 217, 259, 285, 303, 353, 416, 457, 486, 559, 653, 710, 758, 858, 992, 1073, 1148, 1284, 1468, 1591, 1693, 1881, 2128, 2296, 2438, 2694, 3018, 3251, 3455, 3783, 4218, 4522
Offset: 45

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008966, A098236, A307835, A326443, A341068, A341073, A341074, A341075, A341095, 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, 8):
seq(a(n), n=45..105);  # 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, 8];
Table[a[n], {n, 45, 105}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

A341097 Number of partitions of n into 9 distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 3, 3, 7, 7, 8, 10, 16, 18, 18, 22, 33, 35, 39, 47, 65, 69, 77, 89, 117, 126, 138, 163, 205, 223, 242, 282, 344, 376, 407, 466, 561, 612, 664, 751, 889, 966, 1047, 1176, 1365, 1488, 1606, 1792, 2056, 2240, 2406, 2672, 3032, 3286, 3532, 3891
Offset: 58

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008966, A098236, A307835, A326522, A341069, A341073, A341074, A341075, A341095, A341096, 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, 9):
seq(a(n), n=58..113);  # 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, 9];
Table[a[n], {n, 58, 113}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

A341098 Number of partitions of n into 10 distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 6, 8, 7, 10, 14, 17, 17, 22, 32, 35, 37, 47, 62, 71, 72, 91, 114, 132, 136, 167, 205, 234, 247, 293, 355, 398, 426, 497, 590, 661, 708, 819, 956, 1066, 1141, 1306, 1501, 1672, 1791, 2030, 2318, 2559, 2747, 3081, 3490, 3835, 4115
Offset: 72

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008966, A098236, A307835, A326626, A341070, A341073, A341074, A341075, A341095, A341096, A341097.

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, 10):
seq(a(n), n=72..126);  # 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, 10];
Table[a[n], {n, 72, 126}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

A358023 Number of partitions of n into at most 2 distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 3, 4, 5, 5, 4, 4, 5, 5, 5, 5, 7, 5, 5, 5, 6, 6, 5, 6, 8, 7, 7, 7, 11, 8, 7, 8, 11, 9, 8, 10, 12, 10, 8, 9, 13, 10, 8, 8, 13, 11, 10, 8, 13, 11, 11, 10, 14, 12, 11, 11, 15, 12, 11, 12, 17, 13, 13, 12, 21, 14, 14, 13, 19, 15, 13, 15, 20
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 25 2022

Keywords

Crossrefs

Cf. A005117, A087188, A098236, A347648, A347777, A358024, A358025.
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