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A322452 Number of factorizations of n into factors > 1 not including any prime powers.

%I A322452 #11 Feb 09 2019 23:27:10
%S A322452 1,0,0,0,0,1,0,0,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,
%T A322452 1,2,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,1,1,1,1,0,2,0,1,1,0,1,1,0,1,
%U A322452 1,1,0,2,0,1,1,1,1,1,0,1,0,1,0,2,1,1,1,1,0,2,1,1,1,1,1,1,0,1,1,2,0,1,0,1,1
%N A322452 Number of factorizations of n into factors > 1 not including any prime powers.
%C A322452 Also the number of multiset partitions of the multiset of prime indices of n with no constant parts.
%H A322452 Antti Karttunen, <a href="/A322452/b322452.txt">Table of n, a(n) for n = 1..20160</a>
%H A322452 Antti Karttunen, <a href="/A322452/a322452.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H A322452 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%e A322452 The a(840) = 11 factorizations are (6*10*14), (6*140), (10*84), (12*70), (14*60), (15*56), (20*42), (21*40), (24*35), (28*30), (840).
%t A322452 acfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[acfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimePowerQ[#]&]}]];
%t A322452 Table[Length[acfacs[n]],{n,100}]
%o A322452 (PARI) A322452(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(1<omega(d)), s += A322452(n/d, d))); (s)); \\ _Antti Karttunen_, Jan 03 2019
%o A322452 (PARI) first(n) = my(res=vector(n)); for(i=1, n, f=factor(i); v=vecsort(f[,2] , , 4); f[, 2] = v; fb = factorback(f); if(fb==i, res[i] = A322452(i), res[i] = res[fb])); res \\ A322452 the function above \\ _David A. Corneth_, Jan 03 2019
%Y A322452 Positions of 0's are the prime powers A000961.
%Y A322452 Cf. A000688, A001055, A001597, A023893, A023894, A050336, A064573, A294068, A321407, A321760.
%K A322452 nonn
%O A322452 1,36
%A A322452 _Gus Wiseman_, Dec 09 2018
%E A322452 More terms from _Antti Karttunen_, Jan 03 2019