A305436 - OEIS

%I #13 Dec 31 2023 03:40:41

%S 0,1,1,1,1,2,0,1,2,2,0,2,0,1,2,1,1,3,0,2,1,1,0,2,1,1,2,1,0,3,0,1,2,2,

%T 1,3,0,1,1,2,0,2,0,1,3,1,0,2,0,2,2,1,0,3,1,1,1,1,0,3,0,1,2,1,2,3,0,2,

%U 1,2,0,3,0,1,2,1,0,2,0,2,2,1,0,2,2,1,1,1,0,4,0,1,1,1,1,2,0,1,3,2,0,3,0,1,2

%N Number of divisors of n of the form 2^k + 1 for k >= 0.

%C a(n) is the number of terms of A000051 that divide n.

%H Antti Karttunen, <a href="/A305436/b305436.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = Sum_{d|n} A209229(d-1).

%F a(n) = A305435(n) + A209229(n-1).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A323482 = 1.264499... . - _Amiram Eldar_, Dec 31 2023

%t Table[DivisorSum[n, 1 &, IntegerQ@ Log2[# - 1] &], {n, 105}] (* _Michael De Vlieger_, Jun 11 2018 *)

%o (PARI)

%o A209229(n) = (n && !bitand(n,n-1));

%o A305436(n) = sumdiv(n,d,A209229(d-1));

%Y Cf. A000051, A209229, A292315 (positions of zeros), A305435, A323482.

%Y Cf. also A154402.

%K nonn

%O 1,6

%A _Antti Karttunen_, Jun 11 2018