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%I #4 Jan 08 2020 09:45:29
%S 1,2,3,4,6,15,44
%N Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The inverse prime shadow of n is the least number whose prime exponents are the prime indices of n.
%F A001055(A181821(a(n))) = 2^k for some k >= 0.
%e The factorizations of A181821(n) for n = 1, 2, 3, 4, 6, 15:
%e () (2) (4) (6) (12) (72)
%e (2*2) (2*3) (2*6) (8*9)
%e (3*4) (2*36)
%e (2*2*3) (3*24)
%e (4*18)
%e (6*12)
%e (2*4*9)
%e (2*6*6)
%e (3*3*8)
%e (3*4*6)
%e (2*2*18)
%e (2*3*12)
%e (2*2*2*9)
%e (2*2*3*6)
%e (2*3*3*4)
%e (2*2*2*3*3)
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t Select[Range[100],IntegerQ[Log[2,Length[facs[Times@@Prime/@nrmptn[#]]]]]&]
%Y The same for prime numbers (instead of powers of 2) is A330993,
%Y Factorizations are A001055, with image A045782.
%Y Numbers whose number of factorizations is a power of 2 are A330977.
%Y The least number with exactly 2^n factorizations is A330989.
%Y Cf. A033833, A045778, A045783, A181821, A305936, A318283, A318284, A330972, A330973, A330976, A330998, A331022.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Jan 07 2020