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.

A112345 Number of partitions of n into distinct perfect powers.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Sep 05 2005

Keywords

Examples

			a(40) = #{36+4, 32+8, 27+9+4} = 3.

Links

Crossrefs

Cf. A001597, A000009, A033461, A039966, A112344, A284171.

Formula

G.f.: Product_{k>=2} (1 + x^A001597(k)). - Ilya Gutkovskiy, Mar 21 2017

Extensions

a(0)=1 prepended by Ilya Gutkovskiy, Mar 21 2017

A286320 Number of partitions of n into distinct powerful parts (A001694).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 2, 3, 2, 1, 2, 3, 3, 2, 4, 5, 3, 2, 4, 5, 3, 2, 4, 6, 4, 2, 4, 7, 5, 2, 5, 8, 5, 2, 5, 8, 6, 3, 5, 10, 8, 4, 6, 10, 8, 4, 6, 10, 9, 5, 8, 12, 10, 6, 9, 13, 10, 6, 9, 15, 12, 7, 10, 17, 14, 7, 11, 18, 15, 8, 11, 18, 16, 9, 11, 20, 18, 10, 13
Offset: 0

Views

Author

Ilya Gutkovskiy, May 12 2017

Keywords

Examples

			a(25) = 3 because we have [25], [16, 9] and [16, 8, 1].

Links

Crossrefs

Cf. A001694, A284171, A286305.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[(1 + x) Product[1 + Boole[Min@ FactorInteger[k][[All, 2]] > 1] x^k, {k, 2, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A001694(k)).
a(n) = A284171(n) for n < 72.

A331923 Number of compositions (ordered partitions) of n into distinct perfect powers.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 1, 3, 2, 0, 2, 8, 6, 0, 1, 4, 6, 0, 2, 12, 24, 0, 2, 9, 8, 1, 8, 32, 30, 2, 7, 10, 32, 8, 11, 44, 150, 30, 34, 40, 18, 26, 20, 68, 78, 126, 56, 169, 80, 30, 40, 116, 294, 144, 162, 226, 182, 128, 66, 338, 348, 752, 199, 1048
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2020

Keywords

Examples

			a(17) = 4 because we have [16, 1], [9, 8], [8, 9] and [1, 16].

Links

Crossrefs

Cf. A001597, A078635, A112345, A282500, A284171, A331844, A331845.

Programs

  • Maple
    N:= 200: # for a(0)..a(N)
PP:= {1,seq(seq(b^i,i=2..floor(log[b](N))),b=2..floor(sqrt(N)))}:
G:= mul(1+t*x^p, p=PP):
F:= proc(n) local R, k, v;
  R:= normal(coeff(G, x, n));
  add(k!*coeff(R, t, k), k=1..degree(R, t))
end proc:
F(0):= 1:
map(F, [$0..N]); # Robert Israel, Feb 03 2020
  • Mathematica
    M = 200;
PP = Join[{1}, Table[Table[b^i, {i, 2, Floor[Log[b, M]]}], {b, 2, Floor[ Sqrt[M]]}] // Flatten // Union];
G = Product[1 + t x^p, {p, PP}];
a[n_] := Module[{R, k, v}, R = SeriesCoefficient[G, {x, 0, n}]; Sum[k! SeriesCoefficient[R, {t, 0, k}], {k, 1, Exponent[R, t]}]];
a[0] = 1;
a /@ Range[0, M] (* Jean-François Alcover, Oct 25 2020, after Robert Israel *)
Showing 1-3 of 3 results.

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