A366992 The sum of divisors of n that are not terms of A322448.

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 47, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 47, 84, 144

Offset: 1

Author

Amiram Eldar, Oct 31 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A365682 at n = 64.

The sum of divisors of n whose prime factorization has exponents that are all either 1 or primes.

The number of these divisors is A366991(n) and the largest of them is A366994(n).

Links

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

Crossrefs

Cf. A000203, A046100, A365682, A366991, A366994.

Programs

Mathematica

f[p_, e_] := 1 + p + Total[p^Select[Range[e], PrimeQ]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1] + sum(j = 1, f[i, 2], if(isprime(j), f[i, 1]^j)));}

Formula

Multiplicative with a(p^e) = 1 + p + Sum_{primes q <= e} p^q.

a(n) <= A000203(n), with equality if and only if n is a biquadratefree number (A046100).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} f(1/p) = 0.77864544487983775708..., where f(x) = (1-x) * (1 + Sum_{k>=1} (1 + 1/x + Sum_{primes q <= k} 1/x^q) * x^(2*k)).