LINKS
Amiram Eldar, <a href="/A369717/b369717_1.txt">Table of n, a(n) for n = 1..10000</a>
The OEIS is supported by the many generous donors to the OEIS Foundation.
Amiram Eldar, <a href="/A369717/b369717_1.txt">Table of n, a(n) for n = 1..10000</a>
reviewed
approved
proposed
reviewed
editing
proposed
Amiram Eldar, <a href="/A369717/b369717_1.txt">Table of n, a(n) for n = 1..10000</a>
allocated for Amiram EldarThe sum of divisors of the smallest powerful number that is a multiple of n.
1, 7, 13, 7, 31, 91, 57, 15, 13, 217, 133, 91, 183, 399, 403, 31, 307, 91, 381, 217, 741, 931, 553, 195, 31, 1281, 40, 399, 871, 2821, 993, 63, 1729, 2149, 1767, 91, 1407, 2667, 2379, 465, 1723, 5187, 1893, 931, 403, 3871, 2257, 403, 57, 217, 3991, 1281, 2863
1,2
Multiplicative with a(p) = p^2 + p + 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 2.
a(n) >= A000203(n), with equality if and only if n is powerful (A001694).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(3*s-3)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^6 - 1/p^7) = 1.01304866467771286896... .
f[p_, e_] := If[e == 1, p^2 + p + 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
(PARI) a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 1, f[i, 2] = 2)); sigma(f); }
allocated
nonn,easy,mult
Amiram Eldar, Jan 30 2024
approved
editing
allocated for Amiram Eldar
allocated
approved