A389457 The sum of divisors of n that are terms in A048103 (that are not divisible by p^p for any prime p).

1, 3, 4, 3, 6, 12, 8, 3, 13, 18, 12, 12, 14, 24, 24, 3, 18, 39, 20, 18, 32, 36, 24, 12, 31, 42, 13, 24, 30, 72, 32, 3, 48, 54, 48, 39, 38, 60, 56, 18, 42, 96, 44, 36, 78, 72, 48, 12, 57, 93, 72, 42, 54, 39, 72, 24, 80, 90, 60, 72, 62, 96, 104, 3, 84, 144, 68, 54, 96, 144, 72, 39, 74, 114, 124, 60, 96, 168, 80, 18

Offset

1,2

Comments

The number of these divisors is A389456(n), and the largest of them is A327937(n).

Differs from A377517 first at n=625, where a(625)=781, while A377517(625)=156.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

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

a(n) = Sum_{d|n} d*A359550(d).

a(n) = A000203(A327937(n)).

Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Product_{p prime} (1 - 1/p^p) = 0.72199023441955150104... . - Amiram Eldar, Nov 05 2025

Mathematica

f[p_, e_] := (p^(1+Min[p-1, e]) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 05 2025 *)

Prog

(PARI)
A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };
A389457(n) = sumdiv(n, d, d*A359550(d));
(PARI) A389457(n) = { my(f = factor(n)); prod(i = 1, #f~, ((f[i, 1]^(1+min(f[i, 1]-1, f[i, 2])))-1) / (f[i, 1]-1)); };

Crossrefs

Cf. A000203, A048103, A327937, A359550, A389457.

Cf. also A377517, A377520 (variants).

Sequence in context: A183100 A340323 A377517 * A046897 A109506 A369889

Adjacent sequences: A389454 A389455 A389456 * A389458 A389459 A389460

Keyword

nonn,mult,easy

Author

Antti Karttunen, Oct 04 2025

Status

approved