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A114374 Number of partitions of n into parts that are not squarefree.

%I A114374 #16 Dec 31 2016 01:29:09
%S A114374 1,0,0,0,1,0,0,0,2,1,0,0,3,1,0,0,5,2,2,0,7,3,2,0,11,6,4,3,15,8,6,3,22,
%T A114374 13,11,6,34,18,15,9,46,27,24,17,64,43,33,23,89,60,51,37,124,84,78,51,
%U A114374 166,119,109,78,226,168,152,118,300,228,215,166,404,313,300,230,546,421,409
%N A114374 Number of partitions of n into parts that are not squarefree.
%C A114374 a(A078135(n)) = 0; a(A078137(n)) > 0.
%H A114374 Alois P. Heinz, <a href="/A114374/b114374.txt">Table of n, a(n) for n = 0..10000</a>
%F A114374 a(n) = A000041(n) - A073576(n) - A117395(n). - _Reinhard Zumkeller_, Mar 11 2006
%F A114374 G.f.: Product_{k>=1} (1 - mu(k)^2*x^k)/(1 - x^k), where mu(k) is the Moebius function (A008683). - _Ilya Gutkovskiy_, Dec 30 2016
%e A114374 a(12) = #{2*2*3, 2*2*2 + 2*2, 2*2 + 2*2 + 2*2} = 3;
%e A114374 a(13) = #{3*3 + 2*2} = 1.
%p A114374 with(numtheory):
%p A114374 b:= proc(n, i) option remember;
%p A114374       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
%p A114374       `if`(i>n or issqrfree(i), 0, b(n-i, i))))
%p A114374     end:
%p A114374 a:= n-> b(n$2):
%p A114374 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jun 03 2015
%t A114374 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n || SquareFreeQ[i], 0, b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jun 30 2015, after _Alois P. Heinz_ *)
%o A114374 (Haskell)
%o A114374 a114374 = p a013929_list where
%o A114374    p _          0 = 1
%o A114374    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A114374 -- _Reinhard Zumkeller_, Jun 01 2015
%Y A114374 Cf. A013929, A073576.
%Y A114374 Cf. A256012.
%K A114374 nonn
%O A114374 0,9
%A A114374 _Reinhard Zumkeller_, Feb 09 2006
%E A114374 Offset changed and a(0)=1 prepended by _Reinhard Zumkeller_, Jun 01 2015