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

1,2

Links

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

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).

Mathematica

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

Prog

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

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).

Sequence in context: A016475 A037227 A056753 * A396262 A387975 A385073

Adjacent sequences: A385044 A385045 A385046 * A385048 A385049 A385050

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jun 16 2025

Status

approved