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

1,2

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

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)).

Mathematica

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

Prog

(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))); }

Crossrefs

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

Sequence in context: A140782 A284587 A097011 * A365682 A074847 A365170

Adjacent sequences: A366989 A366990 A366991 * A366993 A366994 A366995

Keyword

nonn,easy,mult

Author

Amiram Eldar, Oct 31 2023

Status

approved