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

1,36

Comments

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

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

Example

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).

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}]

Prog

(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
(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

Crossrefs

Positions of 0's are the prime powers A000961.

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

Sequence in context: A352822 A090418 A363877 * A076754 A346482 A364043

Adjacent sequences: A322449 A322450 A322451 * A322453 A322454 A322455

Keyword

nonn

Author

Gus Wiseman, Dec 09 2018

Extensions

More terms from Antti Karttunen, Jan 03 2019

Status

approved