A366744 The sum of divisors of the least coreful infinitary divisor of n.

1, 3, 4, 7, 6, 12, 8, 3, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 12, 31, 42, 4, 56, 30, 72, 32, 3, 48, 54, 48, 91, 38, 60, 56, 18, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 12, 72, 24, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68

Offset: 1

Author

Amiram Eldar, Oct 19 2023

Keywords

• nonn
• easy
• mult

Comments

The number of divisors of the least coreful infinitary divisor of n is A366742(n).

Links

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

Crossrefs

Cf. A000203, A006519, A138302, A339597, A365296, A366742, A366743.

Programs

Mathematica

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

PARI

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

Formula

a(n) = A000203(A365296(n)).

a(n) = A000203(n) if and only if n is in A138302.

Multiplicative with a(p^e) = (p^(A006519(e)+1) - 1)/(p - 1).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p/(p^2-1) + Sum_{e>=1} 1/p^f(e)) = 0.696427154..., where f(k) = 2*k - A006519(k) = A339597(k-1).