A379012 a(n) = A034444(n) if n is powerful (A001694), and 0 otherwise.

1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Kampamolla Venkatasubbareddy and Ayyadurai Sankaranarayanan, On the distribution of the restricted sequence of integers 2^omega(n), INTEGERS, Vol. 24 (2024), #A110.

Formula

a(n) = A112526(n) * A034444(n) = A112526(n) * 2^A001221(n).

Multiplicative with a(p^e) = 2 if e > 1, and 0 otherwise.

Dirichlet g.f.: Product_{p prime} (1 + 2 * Sum_{k>=2} 1/p^(k*s)).

Sum_{k=1..n} a(k) = c_1 * sqrt(n) * log(n) + c_2 * sqrt(n) + c_3 * n^(1/3) * log(n) + c_4 * n^(1/3) + O(n^(1/4) * exp(c*eps*(log(n)/log(log(n)))^(1/3))), for eps > 0 and some c > 0 where c_1, c_2, c_3 and c_4 are constants (Venkatasubbareddy and Sankaranarayanan, 2024).

Mathematica

f[p_, e_] := If[e > 1, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] > 1, 2, 0)); }

Crossrefs

Cf. A001221, A001694, A034444, A112526.

Sequence in context: A100699 A108921 A071548 * A369309 A216228 A372602

Adjacent sequences: A379009 A379010 A379011 * A379013 A379016 A379018

Keyword

nonn,easy,mult,new

Author

Amiram Eldar, Dec 13 2024

Status

approved