A372692 The sum of infinitary divisors of the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 15, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 15, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 15, 40, 1, 1, 3, 1, 1

Offset: 1

Author

Amiram Eldar, May 10 2024

Keywords

• nonn
• easy
• mult

Comments

The sum of divisors d of n that are infinitarily relatively prime to n (see A064379), i.e., d have no common infinitary divisors with n.

The numbers of these divisors is A372331(n) and the largest of them is A372328(n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001146, A036537, A049417, A064379, A372328, A372331.

Programs

Mathematica

f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], ?(# == 0 &)])); a[1] = 1; a[n] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) + 1); Array[a, 100]

PARI

s(n) = apply(x -> 1 - x, binary(n));
a(n) = {my(f = factor(n), k); prod(i = 1, #f~, k = s(f[i, 2]); prod(j = 1, #k, if(k[j], f[i, 1]^(2^(#k-j)) + 1, 1)));}

Formula

Multiplicative with a(p^e) = Product_{k >= 0, 2^k < e, bitand(e, 2^k) = 0} (p^(2^k) + 1).

a(n) >= 1, with equality if and only if n is in A036537.

a(n) <= n-1, with equality if and only if n = 2^(2^k) for k >= 0.