A100007 Number of unitary divisors of 2n-1 (d such that d divides 2n-1, GCD(d,(2n-1)/d)=1). Bisection of A034444.

1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 8, 2, 2, 4, 2, 4, 4, 4, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 2, 4, 4, 2, 8, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 4, 4

Offset

1,2

Links

Table of n, a(n) for n=1..105.

Formula

From Ilya Gutkovskiy, Apr 28 2017: (Start)

a(n) = [x^(2*n-1)] Sum_{k>=1} mu(k)^2*x^k/(1 - x^k).

a(n) = 2^omega(2*n-1). (End)

From Amiram Eldar, Jan 28 2023: (Start)

a(n) = A034444(2*n-1) = A068068(2*n-1).

Sum_{k=1..n} a(k) ~ 4*n*((log(n) + 2*gamma - 1 + 7*log(2)/3 - 2*zeta'(2)/zeta(2)) / Pi^2, where gamma is Euler's constant (A001620). (End)

Example

a(13)=2 because among the three divisors of 25 only 1 and 25 are unitary.

Maple

with(numtheory): for n from 1 to 120 do printf(`%d, `, 2^nops(ifactors(2*n-1)[2])) od: # Emeric Deutsch, Dec 24 2004

Mathematica

a[n_] := 2^PrimeNu[2*n-1]; Array[a, 100] (* Amiram Eldar, Jan 28 2023 *)

Prog

(PARI) a(n) = 2^omega(2*n-1); \\ Amiram Eldar, Jan 28 2023

Crossrefs

Cf. A001620, A034444, A068068, A100008.

Sequence in context: A083261 A003648 A003642 * A104369 A051702 A073130

Adjacent sequences: A100004 A100005 A100006 * A100008 A100009 A100010

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 20 2004

Extensions

More terms from Emeric Deutsch, Dec 24 2004

Status

approved