A100007 - OEIS

%I #16 Jan 28 2023 12:14:25

%S 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,

%T 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,

%U 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

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

%F From _Ilya Gutkovskiy_, Apr 28 2017: (Start)

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

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

%F From _Amiram Eldar_, Jan 28 2023: (Start)

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

%F 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)

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

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

%t a[n_] := 2^PrimeNu[2*n-1]; Array[a, 100] (* _Amiram Eldar_, Jan 28 2023 *)

%o (PARI) a(n) = 2^omega(2*n-1); \\ _Amiram Eldar_, Jan 28 2023

%Y Cf. A001620, A034444, A068068, A100008.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_, Nov 20 2004

%E More terms from _Emeric Deutsch_, Dec 24 2004