cp's OEIS Frontend

A225245 Number of partitions of n into distinct squarefree divisors of n.

%I A225245 #30 Feb 17 2024 02:45:48
%S A225245 1,1,1,1,0,1,2,1,0,0,1,1,1,1,1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,4,1,0,1,
%T A225245 1,1,0,1,1,1,0,1,3,1,0,0,1,1,0,0,0,1,0,1,0,1,0,1,1,1,3,1,1,0,0,1,3,1,
%U A225245 0,1,1,1,0,1,1,0,0,1,2,1,0,0,1,1,2,1,1
%N A225245 Number of partitions of n into distinct squarefree divisors of n.
%C A225245 a(n) <= A033630(n);
%C A225245 a(n) = A033630(n) iff n is squarefree: a(A005117(n)) = A033630(A005117(n));
%C A225245 a(A225353(n)) = 0; a(A225354(n)) > 0.
%H A225245 Alois P. Heinz, <a href="/A225245/b225245.txt">Table of n, a(n) for n = 0..10000</a> (5000 terms from Reinhard Zumkeller)
%H A225245 Noah Lebowitz-Lockard and Joseph Vandehey, <a href="https://arxiv.org/abs/2402.08119">On the number of partitions of a number into distinct divisors</a>, arXiv:2402.08119 [math.NT], 2024. See p. 2.
%F A225245 a(n) = [x^n] Product_{d|n, mu(d) != 0} (1 + x^d), where mu() is the Moebius function (A008683). - _Ilya Gutkovskiy_, Jul 26 2017
%e A225245 a(2*3)     = a(6)  = #{6, 3+2+1} = 2;
%e A225245 a(2*2*3)   = a(12) = #{6+3+2+1} = 1;
%e A225245 a(2*3*5)   = a(30) = #{30, 15+10+5, 15+10+3+2, 15+6+5+3+1} = 4;
%e A225245 a(2*2*3*5) = a(60) = #{30+15+10+5, 30+15+10+3+2, 30+15+6+5+3+1} = 3;
%e A225245 a(2*3*7)   = a(42) = #{42, 21+14+7, 21+14+6+1} = 3;
%e A225245 a(2*2*3*7) = a(84) = #{42+21+14+7, 42+21+14+6+1} = 2.
%t A225245 a[n_] := If[n == 0, 1, Coefficient[Product[If[MoebiusMu[d] != 0, 1+x^d, 1], {d, Divisors[n]}], x, n]];
%t A225245 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Nov 08 2021, after _Ilya Gutkovskiy_ *)
%o A225245 (Haskell)
%o A225245 a225245 n = p (a206778_row n) n where
%o A225245    p _      0 = 1
%o A225245    p []     _ = 0
%o A225245    p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
%Y A225245 Cf. A005117, A008683, A033630, A206778, A008966, A225244, A087188, A225353, A225354.
%K A225245 nonn
%O A225245 0,7
%A A225245 _Reinhard Zumkeller_, May 05 2013