A369761 The sum of divisors of the smallest multiple of n whose prime factorization exponents are all powers of 2.

1, 3, 4, 7, 6, 12, 8, 31, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 124, 31, 42, 121, 56, 30, 72, 32, 511, 48, 54, 48, 91, 38, 60, 56, 186, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 363, 72, 248, 80, 90, 60, 168, 62, 96, 104, 511, 84

Offset: 1

Author

Amiram Eldar, Jan 31 2024

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000203, A138302, A353897, A356194, A369760, A369762.

Programs

Mathematica

f[p_, e_] := (p^(2^Ceiling[Log2[e]]+1)-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

s(n) = {my(e=logint(n, 2)); if(n == 2^e, n, 2^(e+1))};
a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(s(f[i,2])+1)-1)/(f[i,1]-1));}

Formula

a(n) = A000203(A356194(n)).

Multiplicative with a(p^e) = (p^(2^ceiling(log_2(e))+1)-1)/(p-1).

a(n) >= A000203(n), with equality if and only if n is in A138302.