cp's OEIS Frontend

A087188 Number of partitions of n into distinct squarefree parts.

%I A087188 #26 Mar 31 2018 06:36:50
%S A087188 1,1,1,2,1,2,3,3,4,4,5,6,6,8,9,10,13,14,16,18,20,24,27,30,35,37,42,47,
%T A087188 51,59,64,72,81,88,98,109,120,134,147,163,179,195,216,236,258,284,310,
%U A087188 339,371,403,441,480,523,572,621,675,734,796,865,937,1014,1100,1189
%N A087188 Number of partitions of n into distinct squarefree parts.
%H A087188 Alois P. Heinz, <a href="/A087188/b087188.txt">Table of n, a(n) for n = 0..10000</a>
%F A087188 O.g.f.: product_{i=1,2,...infinity} [1+x^A005117(i)]. - _R. J. Mathar_, May 16 2008
%F A087188 a(n) ~ exp(sqrt(2*n)) / (2^(1/4) * sqrt(Pi) * n^(3/4)). - _Vaclav Kotesovec_, Mar 24 2018
%e A087188 n=9: 5+3+1 = 6+2+1 = 6+3 = 7+2: a(9)=4;
%e A087188 n=10: 5+3+2 = 6+3+1 = 7+2+1 = 7+3 = 10: a(10)=5.
%p A087188 with(numtheory):
%p A087188 b:= proc(n, i) option remember;
%p A087188       `if`(i*(i+1)/2<n, 0, `if`(n=0, 1, b(n, i-1)+
%p A087188       `if`(i<=n and issqrfree(i), b(n-i, i-1), 0)))
%p A087188     end:
%p A087188 a:= n-> b(n$2):
%p A087188 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jun 02 2015
%t A087188 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], b[n-i, i-1], 0]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jun 24 2015, after _Alois P. Heinz_ *)
%t A087188 nmax = 100; CoefficientList[Series[Exp[Sum[(-1)^(j + 1)/j * Sum[Abs[MoebiusMu[k]] * x^(j*k), {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 31 2018 *)
%o A087188 (Haskell)
%o A087188 a087188 = p a005117_list where
%o A087188    p _      0 = 1
%o A087188    p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
%o A087188 -- _Reinhard Zumkeller_, Jun 01 2015
%o A087188 (PARI) ok(v)=for(i=2,#v, if(v[i]==v[i-1] || !issquarefree(v[i]), return(0))); #v==0 || issquarefree(v[1])
%o A087188 a(n)=my(s,u); forpart(v=n, if(ok(v), s++)); s \\ _Charles R Greathouse IV_, Nov 05 2017
%Y A087188 Cf. A073576, A000009, A005117, A008966, A225245, A256012.
%K A087188 nonn
%O A087188 0,4
%A A087188 _Reinhard Zumkeller_, Aug 24 2003
%E A087188 Offset changed and a(0)=1 prepended by _Reinhard Zumkeller_, Jun 01 2015