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

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, 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, 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

Offset: 1

Author

Gus Wiseman, Dec 09 2018

Keywords

• nonn

Comments

Also the number of multiset partitions of the multiset of prime indices of n with no constant parts.

Examples

			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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20160
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Positions of 0's are the prime powers A000961.

Cf. A000688, A001055, A001597, A023893, A023894, A050336, A064573, A294068, A321407, A321760.

Programs

Mathematica

acfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[acfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimePowerQ[#]&]}]];
Table[Length[acfacs[n]],{n,100}]

PARI

A322452(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(1A322452(n/d, d))); (s)); \\ Antti Karttunen, Jan 03 2019

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

Extensions

More terms from Antti Karttunen, Jan 03 2019