login
A330990
Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).
7
1, 2, 3, 4, 6, 15, 44
OFFSET
1,2
COMMENTS
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.
FORMULA
A001055(A181821(a(n))) = 2^k for some k >= 0.
EXAMPLE
The factorizations of A181821(n) for n = 1, 2, 3, 4, 6, 15:
() (2) (4) (6) (12) (72)
(2*2) (2*3) (2*6) (8*9)
(3*4) (2*36)
(2*2*3) (3*24)
(4*18)
(6*12)
(2*4*9)
(2*6*6)
(3*3*8)
(3*4*6)
(2*2*18)
(2*3*12)
(2*2*2*9)
(2*2*3*6)
(2*3*3*4)
(2*2*2*3*3)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], IntegerQ[Log[2, Length[facs[Times@@Prime/@nrmptn[#]]]]]&]
CROSSREFS
The same for prime numbers (instead of powers of 2) is A330993,
Factorizations are A001055, with image A045782.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with exactly 2^n factorizations is A330989.
Sequence in context: A056712 A204540 A280592 * A337129 A002087 A049288
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jan 07 2020
STATUS
approved