A354112 Total number of 1's in binary expansion of all divisors of 2^n-1.

1, 3, 4, 9, 6, 17, 8, 27, 16, 33, 20, 100, 14, 44, 42, 81, 18, 186, 20, 293, 80, 118, 38, 634, 62, 77, 64, 523, 80, 813, 32, 243, 153, 99, 154, 5031, 58, 110, 189, 1918, 67, 1624, 115, 1545, 761, 226, 120, 9366, 64, 1728, 472, 1861, 135, 2162, 945, 3471, 261, 1056, 101, 73418

Offset: 1

Author

Michel Marcus, May 17 2022

Keywords

• nonn
• base

Links

• Michel Marcus, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A000225, A093653.

Programs

Maple

a:= n-> add(add(i, i=Bits[Split](d)), d=numtheory[divisors](2^n-1)):
seq(a(n), n=1..60);  # Alois P. Heinz, May 17 2022

Mathematica

a[n_] := Total[DigitCount[Divisors[2^n - 1], 2, 1]]; Array[a, 60] (* Amiram Eldar, May 17 2022 *)

PARI

a(n) = sumdiv(2^n-1, d, hammingweight(d));

Python

# if python version < 3.10, replace d.bitcount() with bin(d).count('1')
from sympy import divisors
def A354112(n): return sum(d.bit_count() for d in divisors(2**n-1,generator=True)) # Chai Wah Wu, May 17 2022

Formula

a(n) = A093653(A000225(n)).