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

1,3

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from indices in prime factorization

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)

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 *)

Prog

(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))));

Crossrefs

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

Cf. also A331309, A331591.

Positions of 1's: A001146.

Sequence in context: A258863 A087207 A179206 * A074987 A294096 A128280

Adjacent sequences: A331737 A331738 A331739 * A331741 A331742 A331743

Keyword

nonn

Author

Antti Karttunen, Feb 05 2020

Status

approved