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A256012 Number of partitions of n into distinct parts that are not squarefree. 4
1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 1, 1, 0, 3, 2, 1, 0, 4, 3, 1, 2, 5, 4, 2, 2, 6, 5, 3, 2, 9, 7, 4, 4, 11, 8, 5, 5, 13, 13, 7, 7, 17, 17, 9, 9, 22, 20, 15, 12, 27, 26, 19, 15, 33, 33, 23, 23, 41, 41, 30, 29, 49, 51, 39, 35, 65, 63, 50, 47, 79 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

Conjecture: a(n) > 0 for n > 23.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

FORMULA

G.f.: Product_{k>=1} (1 + x^k)/(1 + mu(k)^2*x^k), where mu(k) is the Moebius function (A008683). - Ilya Gutkovskiy, Dec 30 2016

EXAMPLE

First nonsquarefree numbers: 4,8,9,12,16,18,20,24,25,27,28, ...  hence

a(20) = #{20, 16+4, 12+8} = 3;

a(21) = #{12+9, 9+8+4} = 2;

a(22) = #{18+4} = 1;

a(23) = #{ } = 0;

a(24) = #{24, 20+4, 16+8, 12+8+4} = 4;

a(25) = #{25, 16+9, 12+9+4} = 3.

MAPLE

with(numtheory):

b:= proc(n, i) option remember;

      `if`(i*(i+1)/2<n, 0, `if`(n=0, 1, b(n, i-1)+

      `if`(i>n or issqrfree(i), 0, b(n-i, i-1))))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..100);  # Alois P. Heinz, Jun 02 2015

MATHEMATICA

b[n_, i_] := b[n, i] = If[i*(i+1)/2<n, 0, If[n==0, 1, b[n, i-1] + If[i>n || SquareFreeQ[i], 0, b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)

PROG

(Haskell)

a256012 = p a013929_list where

   p _      0 = 1

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

CROSSREFS

Cf. A013929, A114374, A087188.

Sequence in context: A249808 A258453 A025874 * A332040 A263251 A318370

Adjacent sequences:  A256009 A256010 A256011 * A256013 A256014 A256015

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Jun 01 2015

STATUS

approved

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Last modified April 4 07:32 EDT 2020. Contains 333213 sequences. (Running on oeis4.)