A331740 Number of prime factors in A225546(n), counted with multiplicity.

0, 1, 2, 1, 4, 3, 8, 2, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 4, 4, 33, 4, 9, 512, 7, 1024, 2, 18, 65, 12, 3, 2048, 129, 34, 6, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 5, 20, 10, 130, 513, 65536, 7, 131072, 1025, 10, 2, 36, 19, 262144, 65, 258, 13, 524288, 4, 1048576, 2049, 6

Offset: 1

Author

Antti Karttunen, Feb 05 2020

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000120, A000720, A001222, A048675, A225546, A331590.

Cf. also A331309, A331591.

Positions of 1's: A001146.

Programs

Mathematica

Array[If[# == 1, 0, PrimeOmega@ Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]] &, 75] (* Michael De Vlieger, Feb 08 2020 *)

PARI

A331740(n) = if(1==n,0,my(f=factor(n)); sum(i=1,#f~,hammingweight(f[i,2])*(2^(primepi(f[i,1])-1))));

Formula

Additive with a(p^e) = A000120(e) * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n).

a(n) = A001222(A225546(n)).

A331591(n) <= a(n) <= A048675(n).

From Peter Munn, Sep 11 2021: (Start)

a(A001146(m)) = 1.

a(A331590(m, k)) = a(m) + a(k).

For squarefree k, a(k*m^2) = a(k) + a(m) = A048675(k) + a(m).

(End)