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 10 results.

A341140 Number of partitions of n into 3 distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 7, 6, 8, 7, 8, 8, 10, 10, 12, 11, 12, 12, 13, 12, 16, 15, 15, 16, 18, 17, 19, 20, 21, 24, 22, 22, 23, 25, 22, 27, 26, 25, 26, 29, 25, 31, 27, 30, 31, 34, 26, 34, 31, 35, 32, 38, 29, 40, 32, 36, 34, 41, 29, 44, 35, 41, 36, 47, 34, 51, 38, 45, 41, 54
Offset: 6

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A282064, A307727, A307825, A341112, A341132, A341141, A341142, A341143, A341144, A341145.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
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`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=6..77);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    q[n_] := q[n] = PrimeNu[n] < 2;
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[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 3];
Table[a[n], {n, 6, 77}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)

A341141 Number of partitions of n into 4 distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 7, 9, 9, 11, 12, 15, 15, 19, 18, 21, 21, 26, 25, 30, 29, 35, 32, 39, 37, 45, 43, 52, 50, 58, 54, 62, 61, 71, 66, 75, 74, 81, 78, 89, 85, 96, 93, 102, 99, 110, 103, 115, 117, 122, 120, 131, 127, 136, 136, 139, 142, 153, 147, 154, 160, 163, 165, 177, 175
Offset: 10

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A341122, A341132, A341133, A341140, A341142, A341143, A341144, A341145.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
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`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 4):
seq(a(n), n=10..76);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    q[n_] := q[n] = PrimeNu[n] < 2;
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[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 4];
Table[a[n], {n, 10, 76}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)

A341142 Number of partitions of n into 5 distinct prime powers (including 1).

Original entry on oeis.org

1, 0, 1, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 15, 18, 21, 23, 26, 31, 33, 36, 41, 43, 48, 52, 58, 62, 72, 72, 82, 85, 95, 97, 112, 112, 125, 127, 142, 142, 161, 159, 181, 180, 200, 196, 222, 217, 243, 239, 269, 261, 291, 284, 316, 308, 341, 332, 370, 358, 394, 381, 427, 414, 456
Offset: 15

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Links

Crossrefs

Cf. A000961, A010055, A341123, A341132, A341134, A341140, A341141, A341143, A341144, A341145.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
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`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 5):
seq(a(n), n=15..78);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    q[n_] := q[n] = PrimeNu[n] < 2;
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[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 5];
Table[a[n], {n, 15, 78}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n,{5}],?(Max[PrimeNu[#]]<2&&Length[#]==Length[Union[#]]&)],{n,15,80}] (* _Harvey P. Dale, Dec 22 2024 *)

A341143 Number of partitions of n into 6 distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 5, 5, 7, 8, 10, 10, 16, 16, 20, 22, 28, 27, 35, 36, 44, 47, 56, 57, 70, 72, 81, 89, 102, 105, 122, 128, 140, 151, 167, 175, 197, 208, 223, 241, 259, 272, 296, 316, 331, 359, 378, 400, 423, 452, 468, 507, 525, 556, 580, 619, 631, 684, 701, 748
Offset: 22

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A341124, A341132, A341135, A341140, A341141, A341142, A341144, A341145.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
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`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 6):
seq(a(n), n=22..81);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    q[n_] := q[n] = Length[FactorInteger[n]] < 2;
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[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 6];
Table[a[n], {n, 22, 81}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341144 Number of partitions of n into 7 distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 6, 8, 8, 10, 12, 15, 16, 22, 23, 29, 30, 37, 40, 50, 50, 63, 68, 79, 81, 99, 101, 121, 127, 147, 153, 182, 182, 214, 224, 253, 262, 304, 309, 351, 365, 405, 421, 477, 485, 541, 563, 614, 634, 706, 719, 791, 823, 888, 919, 1006, 1029, 1115, 1164
Offset: 30

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A341125, A341132, A341136, A341140, A341141, A341142, A341143, A341145.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
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`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 7):
seq(a(n), n=30..90);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    q[n_] := q[n] = Length[FactorInteger[n]] < 2;
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[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 7];
Table[a[n], {n, 30, 90}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341145 Number of partitions of n into 8 distinct prime powers (including 1).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 3, 5, 5, 6, 7, 10, 10, 13, 16, 19, 21, 26, 30, 34, 37, 44, 52, 58, 66, 73, 85, 94, 106, 115, 136, 146, 165, 178, 204, 215, 248, 263, 298, 318, 356, 372, 426, 443, 494, 520, 585, 603, 681, 702, 781, 815, 906, 929, 1044, 1071, 1178, 1223
Offset: 39

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A341126, A341132, A341137, A341140, A341141, A341142, A341143, A341144.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
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`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 8):
seq(a(n), n=39..97);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    q[n_] := q[n] = Length[FactorInteger[n]] < 2;
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[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 8];
Table[a[n], {n, 39, 97}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341146 Number of partitions of n into 9 distinct prime powers (including 1).

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 2, 1, 4, 4, 5, 5, 8, 7, 11, 11, 16, 16, 21, 20, 30, 30, 36, 40, 51, 53, 63, 67, 82, 89, 105, 111, 133, 143, 163, 176, 203, 218, 246, 267, 301, 324, 357, 389, 431, 471, 512, 555, 607, 660, 710, 773, 835, 906, 969, 1057, 1124, 1224, 1298, 1407, 1494
Offset: 50

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Links

Crossrefs

Cf. A000961, A010055, A341127, A341132, A341138, A341140, A341141, A341142, A341143, A341144, A341145, A341147.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
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`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 9):
seq(a(n), n=50..110);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    q[n_] := q[n] = Length[FactorInteger[n]] < 2;
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[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 9];
Table[a[n], {n, 50, 110}] (* Jean-François Alcover, Feb 27 2022, after Alois P. Heinz *)

A358635 Number of partitions of n into at most 2 distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 4, 3, 3, 4, 4, 4, 4, 5, 4, 3, 3, 5, 4, 4, 5, 5, 4, 6, 4, 7, 5, 6, 4, 7, 3, 5, 4, 6, 4, 6, 4, 6, 5, 5, 3, 8, 4, 7, 4, 6, 3, 8, 3, 7, 4, 5, 3, 8, 4, 6, 4, 7, 3, 9, 3, 8, 5, 7, 3, 10, 4, 7, 6, 7, 3, 9, 3, 9, 5, 6, 5, 11, 3, 8, 4, 7, 4, 12, 4
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 24 2022

Keywords

Crossrefs

Cf. A000961, A106244, A341132, A347643, A347762, A358636, A358637.

A341147 Number of partitions of n into 10 distinct prime powers (including 1).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 1, 3, 3, 3, 4, 7, 5, 9, 10, 12, 10, 17, 17, 23, 24, 29, 32, 44, 40, 53, 57, 71, 71, 91, 90, 113, 117, 141, 148, 181, 181, 217, 231, 268, 276, 327, 340, 397, 412, 472, 493, 571, 590, 671, 710, 794, 831, 934, 981, 1094, 1150, 1271, 1345, 1484, 1556, 1706
Offset: 63

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A341131, A341132, A341139, A341140, A341141, A341142, A341143, A341144, A341145, A341146.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
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`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 10):
seq(a(n), n=63..125);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    q[n_] := q[n] = Length[FactorInteger[n]] < 2;
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[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 10];
Table[a[n], {n, 63, 125}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

A365272 a(n) is the least positive integer that can be expressed as the sum of two distinct prime powers (A000961) in exactly n ways.

Original entry on oeis.org

1, 3, 5, 9, 12, 20, 30, 36, 48, 66, 72, 84, 90, 120, 144, 132, 150, 192, 180, 246, 264, 210, 252, 270, 294, 300, 330, 486, 360, 516, 522, 468, 390, 462, 420, 480, 540, 510, 570, 600, 714, 756, 936, 750, 690, 660, 630, 810, 780, 924, 870, 1296, 930, 1122, 1404, 840
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 07 2023

Keywords

Examples

			For n = 3: 9 = 1 + 8 = 2 + 7 = 4 + 5.

Crossrefs

Cf. A000961, A087747, A225099, A341132.
Showing 1-10 of 10 results.

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