A179051 Number of partitions of n into powers of 10 (cf. A011557).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Author

Reinhard Zumkeller, Jun 27 2010

Keywords

• nonn

Comments

A179052 and A008592 give record values and where they occur.

Examples

			a(19) = #{10 + 9x1, 19x1} = 2;
a(20) = #{10 + 10, 10 + 10x1, 20x1} = 3;
a(21) = #{10 + 10 + 1, 10 + 11x1, 21x1} = 3.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for related partition-counting sequences

Crossrefs

Number of partitions of n into powers of b: A018819 (b=2), A062051 (b=3).

Cf. A206245, A000041, A179051.

Cf. A133880, A132272.

Cf. A008592, A179052.

Programs

Haskell

a179051 = p 1 where
   p _ 0 = 1
   p k m = if m < k then 0 else p k (m - k) + p (k * 10) m
-- Reinhard Zumkeller, Feb 05 2012

Mathematica

terms = 10001;
CoefficientList[Product[1/(1 - x^(10^k)) + O[x]^terms,
     {k, 0, Log[10, terms] // Ceiling}], x]
(* Jean-François Alcover, Dec 12 2021, after Ilya Gutkovskiy *)

Formula

a(n) = A133880(n) for n < 90; a(n) = A132272(n) for n < 100.

a(10^n) = A145513(n).

a(10*n) = A179052(n).

A179052(n) = a(A008592(n));

a(n) = p(n,1) where p(n,k) = if k<=n then p(10*[(n-k)/10],k)+p(n,10*k) else 0^n.

G.f.: Product_{k>=0} 1/(1 - x^(10^k)). - Ilya Gutkovskiy, Jul 26 2017