A147583 Number of partitions of n into distinct parts such that 5*u<=v for all pairs (u,v) of parts with u<v.

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 12, 13, 13, 13, 13, 13, 14, 15, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 19, 20, 21, 21, 21, 21, 22, 23, 24, 24, 24, 24, 25, 26, 27, 27, 27, 27

Offset

0,7

Comments

From Edward Early, Jan 10 2009: (Start)

Also the dimension of the n-th degree part of the mod 5 Steenrod algebra.

Also the number of partitions into parts (5^j-1)/4=1+5+5^2+...+5^(j-1) for j>=1. (End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (terms n = 1..1000 from Reinhard Zumkeller)

Example

a(29) = #{29,28+1,27+2,26+3,25+4} = 5;
a(30) = #{30,29+1,28+2,27+3,26+4,25+5} = 6;
a(31) = #{31,30+1,29+2,28+3,27+4,26+5,25+5+1} = 7.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(p-> `if`(p>n, 0, b(n-p, i)))(5^i-1)))
    end:
a:= n-> b(4*n, 1+ilog[5](4*n)):
seq(a(n), n=0..83);  # Alois P. Heinz, Oct 01 2025

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + Function[p, If[p > n, 0, b[n-p, i]]][5^i-1]]];
a[n_] := b[4n, Length[IntegerDigits[4n, 5]]];
Table[a[n], {n, 0, 83}] (* Jean-François Alcover, Jan 20 2026, after Alois P. Heinz *)

Prog

(Haskell)
a147583 = p [1..] where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p [5 * k ..] (m - k) + p ks m
-- Reinhard Zumkeller, Oct 10 2013

Crossrefs

Cf. A000041, A000009, A000929, A132011.

Sequence in context: A242602 A097992 A195177 * A054895 A194699 A262694

Adjacent sequences: A147580 A147581 A147582 * A147584 A147585 A147586

Keyword

nonn

Author

Reinhard Zumkeller, following a suggestion of Clark Hanley (clark.hanley(AT)gmx.com), Nov 08 2008

Extensions

a(0)=1 prepended by Alois P. Heinz, Oct 01 2025

Status

approved