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 1-3 of 3 results.

A240128 Number of partitions of n such that the sum of cubes of the parts is a cube.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 3, 4, 4, 4, 3, 3, 4, 4, 5, 12, 9, 14, 13, 13, 16, 17, 30, 34, 33, 34, 37, 50, 57, 64, 73, 99, 101, 114, 125, 141, 187, 193, 226, 264, 286, 326, 365, 456, 506, 565, 655, 742, 809, 911, 1071, 1233, 1392, 1506, 1744, 2046
Offset: 0

Views

Author

Clark Kimberling, Apr 02 2014

Keywords

Examples

			a(17) counts these 4 partitions:  [17], [4,3,3,1,1,1,1,1,1,1], [4,3,2,2,2,2,1,1], [3,3,3,3,2,2,1].

Links

Crossrefs

Cf. A240127.

Programs

  • Mathematica
    f[x_] := x^(1/3); z = 26; ColumnForm[t = Map[Select[IntegerPartitions[#], IntegerQ[f[Total[#^3]]] &] &, Range[z]] ](* shows the partitions *)
t2 = Map[Length[Select[IntegerPartitions[#], IntegerQ[f[Total[#^2]]] &]] &, Range[40]] (* A240128 *) (* Peter J. C. Moses, Apr 01 2014 *)
  • PARI
    a(n)=my(s); forpart(v=n, s+=ispower(sum(i=1, #v, v[i]^3),3)); s \\ Charles R Greathouse IV, Mar 06 2017

Extensions

a(0)=1 prepended by Alois P. Heinz, Oct 03 2024

A375731 a(n) is the number of partitions of n having a square number of parts whose sum of squares is a square.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 4, 2, 2, 7, 4, 4, 7, 7, 6, 9, 12, 9, 21, 21, 19, 26, 30, 32, 43, 54, 54, 64, 87, 85, 119, 128, 146, 174, 205, 213, 281, 324, 368, 420, 503, 531, 688, 760, 837, 992, 1174, 1252, 1535, 1705, 1931, 2236, 2619, 2821, 3402, 3769, 4272
Offset: 0

Views

Author

Felix Huber, Aug 28 2024

Keywords

Examples

			a(13) counts the 4 partitions [1, 1, 1, 1, 1, 1, 1, 3, 3] with 9 = 3^2 parts and 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 3^2 + 3^2 = 5^2, [1, 4, 4, 4] with 2^2 parts and 1^2 + 4^2 + 4^2 + 4^2 = 7^2, [2, 2, 4, 5] with 4 = 2^2 parts and 2^2 + 2^2 + 4^2 + 5^2 = 7^2, [13] with 1 = 1^2 part and 13^2 = 13^2.

Crossrefs

Cf. A001156, A025428, A006456, A037444, A240127, A240128, A375732.

Programs

  • Maple
    # first Maple program to calculate the sequence:
A375731:=proc(n) local a,i,j; a:=0; for i in combinat:-partition(n) do if issqr(numelems(i)) and issqr(add(i[j]^2,j=1..nops(i))) then a:=a+1 fi od; return a end proc; seq(A375731(n),n=0..63);
# second Maple program to calculate the partitions:
A375731part:=proc(n) local L,i,j;L:=[]; for i in combinat:-partition(n) do if issqr(numelems(i)) and issqr(add(i[j]^2,j=1..nops(i))) then L:=[op(L),i] fi od; return op(L) end proc; A375731part(13);
  • PARI
    a(n) = my(nb=0); forpart(p=n, if (issquare(#p) && issquare(norml2(Vec(p))), nb++)); nb; \\ Michel Marcus, Aug 30 2024

Formula

1 <= a(n) <= A240127(n).

A375732 a(n) is the number of partitions of n having a cube number of parts whose sum of cubes is a cube.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 5, 2, 2, 1, 2, 2, 2, 4, 5, 9, 4, 5, 2, 6, 9, 9, 13, 12, 16, 8, 10, 8, 13, 19, 20, 26, 23, 23, 22, 22, 30, 38, 45, 47, 60, 54, 77, 87, 83, 89, 88, 104, 131, 156, 170, 202, 208, 220, 241
Offset: 0

Views

Author

Felix Huber, Aug 28 2024

Keywords

Examples

			a(37) counts the 4 partitions [1, 1, 1, 2, 6, 8, 9, 9] with 8 = 2^3 parts and 1^3 + 1^3 + 1^3 + 2^3 + 6^3 + 8^3 + 9^3 + 9^3 = 13^3, [1, 1, 2, 4, 4, 6, 8, 11] with 8 = 2^3 parts and 1^3 + 1^3 + 2^3 + 4^3 + 4^3 + 6^3 + 8^3 + 11^3 = 13^3, [1, 1, 1, 2, 2, 2, 10, 18] with 8 = 2^3 parts and 1^3 + 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 10^3 + 18^3 = 19^3, [37] with 1 = 1^3 part and 37^3 = 37^3.

Crossrefs

Cf. A003108, A025457, A240127, A240128, A259792, A375731.

Programs

  • Maple
    # first Maple program to calculate the sequence:
A375732:=proc(n) local a,i,j; a:=0; for i in combinat:-partition(n) do if type(root(numelems(i),3),integer) and type(root(add(i[j]^3,j=1..nops(i)),3), integer) then a:=a+1 fi od; return a end proc; seq(A375732(n),n=0..75);
# second Maple program to calculate the partitions:
A375732part:=proc(n) local L,i,j; L:=[]; for i in combinat:-partition(n) do if type(root(numelems(i),3),integer) and type(root(add(i[j]^3,j=1..nops(i)), 3),integer) then L:=[op(L),i] fi od; return op(L); end proc; A375732part(37);
  • PARI
    a(n) = my(nb=0); forpart(p=n, if (ispower(#p,3) && ispower(sum(k=1, #p, p[k]^3),3), nb++)); nb; \\ Michel Marcus, Sep 01 2024

Formula

1 <= a(n) <= A240128(n).
Showing 1-3 of 3 results.

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