A339916 The sum of 2^((d-1)/2) over all divisors of 2n+1.

1, 3, 5, 9, 19, 33, 65, 135, 257, 513, 1035, 2049, 4101, 8211, 16385, 32769, 65571, 131085, 262145, 524355, 1048577, 2097153, 4194455, 8388609, 16777225, 33554691, 67108865, 134217765, 268435971, 536870913, 1073741825, 2147484699, 4294967365, 8589934593, 17179871235, 34359738369, 68719476737

Offset: 0

Author

Don Knuth, Dec 22 2020

Keywords

• nonn

Comments

This is sort of a bitmap representation of the divisors of odd numbers.

Examples

			For n=7, a(7)=2^7+2^2+2^1+2^0=135 because the divisors of 15 are 15,5,3,1.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..3318

Crossrefs

Cf. A114001 (bit reversal), A034729, A055895.

Programs

Maple

seq(add(2^((d-1)/2),d=numtheory:-divisors(2*n+1)),n=0..100); # Robert Israel, Dec 24 2020

Mathematica

A339916[n_]:=Block[{d=Divisors[2n+1]},Sum[2^((d[[k]]-1)/2),{k,Length[d]}]];Array[A339916,50,0]

PARI

a(n) = sumdiv(2*n+1, d, 2^((d-1)/2)); \\ Michel Marcus, Dec 23 2020

Python

from sympy import divisors
def a(n): return sum(2**((d-1)//2) for d in divisors(2*n+1))
print([a(n) for n in range(37)]) # Michael S. Branicky, Dec 24 2020