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A322453 Number of factorizations of n into factors > 1 using only primes and perfect powers.

%I A322453 #18 Dec 30 2018 00:04:11
%S A322453 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,5,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,7,1,1,
%T A322453 1,5,1,1,1,3,1,1,1,2,2,1,1,5,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,11,1,1,1,2,
%U A322453 1,1,1,7,1,1,2,2,1,1,1,5,5,1,1,2,1,1,1,3,1,2,1,2,1,1,1,7,1,2,2,5,1,1,1,3,1
%N A322453 Number of factorizations of n into factors > 1 using only primes and perfect powers.
%C A322453 First differs from A000688 at a(36) = 5, A000688(36) = 4.
%C A322453 Terms in this sequence only depend on the prime signature of n. - _David A. Corneth_, Dec 26 2018
%H A322453 Antti Karttunen, <a href="/A322453/b322453.txt">Table of n, a(n) for n = 1..16384</a>
%H A322453 Antti Karttunen, <a href="/A322453/a322453.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H A322453 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%e A322453 The a(144) = 13 factorizations:
%e A322453   (144),
%e A322453   (4*36), (9*16),
%e A322453   (2*2*36), (2*8*9), (3*3*16), (4*4*9),
%e A322453   (2*2*4*9), (2*3*3*8), (3*3*4*4),
%e A322453   (2*2*2*2*9), (2*2*3*3*4),
%e A322453   (2*2*2*2*3*3).
%t A322453 perpowQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1;
%t A322453 pfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[pfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],Or[PrimeQ[#],perpowQ[#]]&]}]];
%t A322453 Table[Length[pfacs[n]],{n,100}]
%o A322453 (PARI) A322453(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(ispower(d)||isprime(d)), s += A322453(n/d, d))); (s)); \\ _Antti Karttunen_, Dec 26 2018
%Y A322453 Cf. A000688, A000961, A001055, A001597, A025487, A050336, A284696, A294068, A320322, A322452.
%K A322453 nonn
%O A322453 1,4
%A A322453 _Gus Wiseman_, Dec 09 2018
%E A322453 More terms from _Antti Karttunen_, Dec 24 2018