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A225244 Number of partitions of n into squarefree divisors of n. 10
1, 1, 2, 2, 3, 2, 8, 2, 5, 4, 11, 2, 27, 2, 14, 14, 9, 2, 64, 2, 40, 18, 20, 2, 125, 6, 23, 10, 53, 2, 742, 2, 17, 26, 29, 26, 343, 2, 32, 30, 195, 2, 1654, 2, 79, 136, 38, 2, 729, 8, 341, 38, 92, 2, 1000, 38, 265, 42, 47, 2, 14188, 2, 50, 184, 33, 44, 5257, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) <= A018818(n);

a(n) = A018818(n) iff n is squarefree: a(A005117(n)) = A018818(A005117(n));

a(A000040(n)) = 2.

LINKS

Reinhard Zumkeller and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (300 terms from Reinhard Zumkeller)

FORMULA

a(n) = [x^n] Product_{d|n, mu(d) != 0} 1/(1 - x^d), where mu() is the Moebius function (A008683). - Ilya Gutkovskiy, Jul 26 2017

EXAMPLE

a(8) = #{2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+6x1, 8x1} = 5;

a(9) = #{3+3+3, 3+3+1+1+1, 3+1+1+1+1+1+1, 9x1} = 4;

a(10) = #{10, 5+5, 5+2+2+1, 5+2+1+1+1, 5+5x1, 2+2+2+2+2, 2+2+2+2+1+1, 2+2+2+1+1+1+1, 2+2+6x1, 2+8x1, 10x1} = 11;

a(11) = #{11, 1+1+1+1+1+1+1+1+1+1+1} = 2;

a(12) = #{6+6, 6+3+3, 6+3+2+1, 6+3+1+1+1, 6+2+2+2, 6+2+2+1+1, 6+2+1+1+1+1, 6+6x1, 3+3+3+3, 3+3+3+2+1, 3+3+3+1+1+1, 3+3+2+2+2, 3+3+2+2+1+1, 3+3+2+4x1, 3+3+6x1, 3+2+2+2+2+1, 3+2+2+2+1+1+1, 3+2+2+5x1, 3+2+7x1, 3+8x1, 2+2+2+2+2+2, 2+2+2+2+2+1+1, 2+2+2+2+1+1+1+1, 2+2+2+6x1, 2+2+8x1, 2+10x1, 12x1} = 27;

a(13) = #{11, 1+1+1+1+1+1+1+1+1+1+1+1+1} = 2;

a(14) = #{14, 7+7, 7+2+2+2+1, 7+2+2+1+1+1, 7+2+5x1, 7+7x1, 7x2, 6x2+1+1, 5x2+1+1+1+1, 4x2+6x1, 2+2+2+8x1, 2+2+10x1, 2+12x1, 14x1} = 14;

a(15) = #{15, 5+5+5, 5+5+3+1+1, 5+5+5x1, 5+3+3+3+1, 5+3+3+1+1+1+1, 5+3+7x1, 5+10x1, 3+3+3+3+3, 3+3+3+3+1+1+1, 3+3+3+6x1, 3+3+9x1, 3+12x1, 15x1} = 14.

MAPLE

with(numtheory):

a:= proc(n) local b, l; l:= sort([select(issqrfree, divisors(n))[]]):

      b:= proc(m, i) option remember; `if`(m=0 or i=1, 1,

            `if`(i<1, 0, b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))

          end; forget(b):

      b(n, nops(l))

    end:

seq(a(n), n=0..100); # Alois P. Heinz, Feb 05 2014

MATHEMATICA

a[0] = 1; a[n_] := Module[{b, l}, l = Select[Divisors[n], SquareFreeQ]; b[m_, i_] := b[m, i] = If[m == 0 || i == 1, 1, If[i < 1, 0, b[m, i - 1] + If[l[[i]] > m, 0, b[m - l[[i]], i]]]]; b[n, Length[l]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 27 2015, after Alois P. Heinz *)

PROG

(Haskell)

a225244 n = p (a206778_row n) n where

   p _          0 = 1

   p []         _ = 0

   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

CROSSREFS

Cf. A206778, A005117, A008966, A225245, A073576.

Sequence in context: A117754 A248577 A015999 * A345281 A338319 A280583

Adjacent sequences:  A225241 A225242 A225243 * A225245 A225246 A225247

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 05 2013

STATUS

approved

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Last modified September 19 07:08 EDT 2021. Contains 347554 sequences. (Running on oeis4.)