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Number of ways of factoring n with all factors greater than 1 (a(1)=1 by convention) minus number of nonprime divisors of n.
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%I #10 Jul 13 2017 21:13:53

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,

%T 0,2,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,1,0,0,0,2,0,0,0,5,0,0,0,0,

%U 0,0,0,6,0,0,0,0,0,0,0,4,1,0,0,2,0,0,0,1,0,2,0,0,0,0,0,9,0,0,0,2,0,0,0,1,0

%N Number of ways of factoring n with all factors greater than 1 (a(1)=1 by convention) minus number of nonprime divisors of n.

%H Antti Karttunen, <a href="/A167688/b167688.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = A001055(n) - A033273(n).

%t c[1, r_] := c[1, r] = 1; c[n_, r_] := c[n, r] = Module[{d, i}, d = Select[Divisors@ n, 1 < # <= r &]; Sum[c[n/d[[i]], d[[i]]], {i, 1, Length@ d}]]; Array[c[#, #] - DivisorSum[#, 1 &, ! PrimeQ@ # &] &, 105] (* _Michael De Vlieger_, Jul 12 2017, after _Dean Hickerson_ at A001055 *)

%o (PARI)

%o fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if((d > 1) && (d <= m), s=s+fcnt(n/d, d)))); s};

%o A001055(n) = fcnt(n, n); \\ This function from _Michael B. Porter_, Oct 29 2009

%o A167688(n) = A001055(n) - (numdiv(n) - omega(n)); \\ _Antti Karttunen_, Jul 12 2017

%Y Cf. A001055, A033273.

%K nonn

%O 1,32

%A _Juri-Stepan Gerasimov_, Nov 09 2009

%E a(64) and a(80) corrected by _R. J. Mathar_, May 30 2010