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.

A219107 Number of compositions (ordered partitions) of n into distinct prime parts.

Original entry on oeis.org

1, 0, 1, 1, 0, 3, 0, 3, 2, 2, 8, 1, 8, 3, 8, 8, 10, 25, 16, 9, 16, 38, 16, 61, 18, 62, 46, 66, 160, 91, 138, 99, 70, 122, 306, 126, 314, 151, 362, 278, 588, 901, 602, 303, 654, 1142, 888, 1759, 892, 1226, 950, 2160, 1230, 3379, 1444, 2372, 2100, 4644, 7416
Offset: 0

Views

Author

Alois P. Heinz, Nov 11 2012

Keywords

Comments

a(0) = 0 iff n in {1,4,6}.

Examples

			a(5) = 3: [2,3], [3,2], [5].
a(7) = 3: [2,5], [5,2], [7].
a(8) = 2: [3,5], [5,3].
a(9) = 2: [2,7], [7,2].
a(10) = 8: [2,3,5], [2,5,3], [3,2,5], [3,5,2], [5,2,3], [5,3,2], [3,7], [7,3].

Links

Crossrefs

Cf. A000142, A000586, A032021, A218396.

Programs

  • Maple
    with(numtheory):
b:= proc(n, i) b(n, i):=
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
a:= proc(n) local l; l:= b(n, pi(n));
      a(n):= add(l[i]*(i-1)!, i=1..nops(l))
    end:
seq(a(n), n=0..70);
# second Maple program:
s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 30 2020
  • Mathematica
    zip = With[{m=Max[Length[#1], Length[#2]]}, PadRight[#1, m]+PadRight[#2, m] ]&;
b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, b[n, i-1] ~zip~ Join[{0}, If[Prime[i] > n, {}, b[n - Prime[i], i-1]]], {0}]];
a[n_] := Module[{l}, l = b[n, PrimePi[n]]; Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Mar 24 2017, adapted from Maple *)

Formula

a(n) = Sum_{k=0..A024936(n)} A219180(n,k)*k!.

A331844 Number of compositions (ordered partitions) of n into distinct squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 3, 8, 0, 0, 8, 30, 0, 0, 0, 2, 6, 1, 2, 6, 24, 2, 8, 6, 0, 0, 8, 30, 0, 0, 7, 32, 24, 2, 8, 30, 120, 6, 24, 2, 6, 0, 8, 36, 24, 1, 34, 150, 0, 2, 12, 30, 24, 0, 2, 38, 150, 0, 12, 78, 144, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(14) = 6 because we have [9,4,1], [9,1,4], [4,9,1], [4,1,9], [1,9,4] and [1,4,9].

Links

Crossrefs

Cf. A000290, A006456, A032020, A032021, A032022, A033461, A218396, A219107, A331843, A331845, A331846, A331847.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(i*(i+1)*(2*i+1)/6n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..82);  # Alois P. Heinz, Jan 30 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n, Sqrt[n] // Floor, 0];
a /@ Range[0, 82] (* Jean-François Alcover, Oct 29 2020, after Alois P. Heinz *)

A331843 Number of compositions (ordered partitions) of n into distinct triangular numbers.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 0, 2, 7, 2, 0, 2, 6, 1, 4, 6, 2, 12, 24, 3, 8, 0, 8, 32, 6, 2, 13, 26, 6, 34, 36, 0, 32, 150, 3, 20, 50, 14, 54, 126, 32, 32, 12, 55, 160, 78, 122, 44, 174, 4, 72, 294, 36, 201, 896, 128, 62, 180, 176, 164, 198, 852, 110, 320, 159, 212, 414
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(10) = 7 because we have [10], [6, 3, 1], [6, 1, 3], [3, 6, 1], [3, 1, 6], [1, 6, 3] and [1, 3, 6].

Links

Crossrefs

Cf. A000217, A023361, A024940, A032020, A032021, A032022, A218396, A219107, A331844, A331845, A331846, A331847.

Programs

  • Maple
    h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), 1+h(n-1), h(n-1)))
    end:
b:= proc(n, i, p) option remember; (t->
      `if`(t*(i+2)/3n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
    end:
a:= n-> b(n, h(n), 0):
seq(a(n), n=0..73);  # Alois P. Heinz, Jan 31 2020
  • Mathematica
    h[n_] := h[n] = If[n<1, 0, If[IntegerQ @ Sqrt[8n+1], 1 + h[n-1], h[n-1]]];
b[n_, i_, p_] := b[n, i, p] = Function[t, If[t (i + 2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t>n, 0, b[n - t, i - 1, p + 1]]]]][(i(i + 1)/2)];
a[n_] := b[n, h[n], 0];
a /@ Range[0, 73] (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

A331845 Number of compositions (ordered partitions) of n into distinct cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 6, 24
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(36) = 6 because we have [27,8,1], [27,1,8], [8,27,1], [8,1,27], [1,27,8] and [1,8,27].

Links

Crossrefs

Cf. A000578, A023358, A032020, A032021, A032022, A218396, A219107, A279329, A331843, A331844, A331846, A331847.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`((i*(i+1)/2)^2n, 0, b(n-i^3, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, iroot(n, 3), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 30 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[(i(i+1)/2)^2 < n, 0, If[n == 0, p!, If[i^3 > n, 0, b[n-i^3, i-1, p+1]] + b[n, i-1, p]]];
a[n_] := b[n, Floor[n^(1/3)], 0];
a /@ Range[0, 100] (* Jean-François Alcover, Oct 31 2020, after Alois P. Heinz *)

A331846 Number of compositions (ordered partitions) of n into distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 9, 5, 12, 16, 21, 41, 42, 49, 59, 79, 130, 231, 230, 295, 226, 495, 609, 699, 1472, 1042, 1377, 2308, 2982, 3425, 3879, 4877, 7156, 7189, 13531, 14797, 13570, 19551, 27667, 30327, 36382, 47519, 60783, 70561, 78330, 136988, 121659, 174851
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(7) = 5 because we have [7], [6, 1], [5, 2], [2, 5] and [1, 6].

Links

Crossrefs

Cf. A005117, A032020, A032021, A032022, A087188, A218396, A219107, A280194, A331843, A331844, A331845, A331847.

A331847 Number of compositions (ordered partitions) of n into distinct prime powers (1 excluded).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 5, 3, 11, 10, 13, 18, 19, 52, 30, 61, 77, 114, 109, 146, 260, 318, 341, 356, 631, 666, 927, 848, 1849, 1978, 2305, 2213, 3560, 4302, 4748, 5588, 6779, 13952, 9044, 15534, 16897, 25084, 20731, 29524, 34882, 49360, 50765, 55112, 106903, 83652, 128552, 106638
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(10) = 10 because we have [8, 2], [7, 3], [5, 3, 2], [5, 2, 3], [3, 7], [3, 5, 2], [3, 2, 5], [2, 8], [2, 5, 3] and [2, 3, 5].

Links

Crossrefs

Cf. A032020, A032021, A032022, A035942, A054685, A218396, A219107, A246655, A280195, A331843, A331844, A331845, A331846.

A331922 Number of compositions (ordered partitions) of n into distinct Lucas numbers (beginning with 1).

Original entry on oeis.org

1, 1, 0, 1, 3, 2, 0, 3, 8, 0, 2, 9, 8, 0, 8, 32, 6, 0, 9, 32, 0, 8, 38, 30, 0, 32, 150, 0, 6, 33, 32, 0, 32, 158, 30, 0, 38, 174, 0, 30, 176, 150, 0, 150, 870, 24, 0, 33, 152, 0, 32, 182, 150, 0, 158, 894, 0, 30, 182, 174, 0, 174, 1014, 144, 0, 176, 990, 0, 150, 1014, 864
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2020

Keywords

Examples

			a(7) = 3 because we have [7], [4, 3] and [3, 4].

Links

Crossrefs

Cf. A000204, A003263, A054770 (positions of 0's), A067592, A067595, A218396, A288039.

A357306 Number of compositions (ordered partitions) of n into distinct Lucas numbers (beginning at 2).

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 8, 9, 8, 8, 32, 9, 14, 32, 38, 32, 36, 150, 33, 32, 32, 158, 38, 60, 174, 176, 150, 150, 870, 33, 56, 152, 182, 158, 180, 894, 182, 174, 174, 1014, 176, 294, 990, 1014, 870, 888, 5904, 153, 152, 152, 902, 182, 300, 1014, 1022, 894, 894
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 23 2022

Keywords

Links

Crossrefs

Cf. A000032, A003263, A067593, A067595, A218396, A288039, A331922.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.