A366992 - OEIS

%I #9 Nov 02 2023 09:15:30

%S 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,

%T 42,40,56,30,72,32,47,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,60,

%U 57,93,72,98,54,120,72,120,80,90,60,168,62,96,104,47,84,144

%N The sum of divisors of n that are not terms of A322448.

%C First differs from A365682 at n = 64.

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

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

%H Amiram Eldar, <a href="/A366992/b366992.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F 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)).

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

%o (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)));}

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

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Oct 31 2023