A076277 Number of product signs needed to write all the factorizations of n with all factors > 1.

0, 0, 0, 1, 0, 1, 0, 3, 1, 1, 0, 4, 0, 1, 1, 7, 0, 4, 0, 4, 1, 1, 0, 10, 1, 1, 3, 4, 0, 5, 0, 13, 1, 1, 1, 13, 0, 1, 1, 10, 0, 5, 0, 4, 4, 1, 0, 22, 1, 4, 1, 4, 0, 10, 1, 10, 1, 1, 0, 16, 0, 1, 4, 24, 1, 5, 0, 4, 1, 5, 0, 30, 0, 1, 4, 4, 1, 5, 0, 22, 7, 1, 0, 16, 1, 1, 1, 10, 0, 16, 1, 4, 1, 1, 1, 42, 0

Offset: 1

Author

Remco van Sabben (pvsabben(AT)stwillibrord.nl), Oct 04 2002

Keywords

• nonn

Examples

			12 = 3*4 = 2*6 = 2*2*3, 4 product signs are needed, so a(12) = 4.
24 = 12*2 = 6*2*2 = 4*3*2 = 3*2*2*2 = 8*3 = 6*4 with 10 multiplies so a(24) = 10.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001055, A066637.

Programs

Mathematica

g[1, r_] := g[1, r]={1, 0}; g[n_, r_] := g[n, r]=Module[{ds, i, val}, ds=Select[Divisors[n], 1<#<=r&]; val={0, 0}+Sum[g[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]; val+{0, val[[1]]}]; a[1]=0; a[n_] := g[n, n][[2]]-g[n, n][[1]]; a/@Range[97] (* g[n, r] = {c, f}, where c is the number of factorizations of n with factors <= r and f is the total number of factors in them. - Dean Hickerson, Oct 10 2002 *)

PARI

A076277(n) = a(n,0);
a(n, k=0) = if(k<=0, a(n, 2)[2], if(n<=1||k>n, [0, 0], [1, 0]+sumdiv(n, d, if(d>=max(2, k)&&d<=n/d, a(n/d, d)*[1, 1; 0, 1], [0, 0])))); \\ From the original author. | and & replaced with || and && to conform with modern PARI-systems. - Antti Karttunen, May 25 2017

Formula

a(n) = A066637(n) - A001055(n) for n > 1. - Henry Bottomley, Oct 10 2002

Extensions

More terms from Robert G. Wilson v and Michael Somos, Oct 08 2002