A385047 The sum of the unitary divisors of n that are powers of 2.

1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 17, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 33, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 17, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 65, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 17, 1, 3, 1, 5, 1, 3

Offset: 1

Author

Amiram Eldar, Jun 16 2025

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

The unitary analog of A038712.

Cf. A001620, A006519, A034448.

The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), this sequence (power of 2), A385048 (cubefull), A385049 (biquadratefree).

Programs

Mathematica

a[n_] := If[OddQ[n], 1, 2^IntegerExponent[n, 2] + 1]; Array[a, 100]

PARI

a(n) = if(n%2, 1, 2^valuation(n, 2)+1);

Formula

Multiplicative with a(2^e) = 2^e + 1, and a(p^e) = 1 for an odd prime p.

a(n) = A034448(n) / A192066(n).

a(n) = A059841(n) + A006519(n), i.e., a(n) = A006519(n) + 1 if n is even, and 1 is n is odd.

Dirichlet g.f.: zeta(s) * ((1-1/2^(2*s-1))/(1-1/2^(s-1))).

Sum_{k=1..n} a(k) ~ (n/(2*log(2))) * (log(n) + gamma - 1 + 5*log(2)/2), where gamma is Euler's constant (A001620).