A369718 The sum of unitary divisors of the smallest powerful number that is a multiple of n.
1, 5, 10, 5, 26, 50, 50, 9, 10, 130, 122, 50, 170, 250, 260, 17, 290, 50, 362, 130, 500, 610, 530, 90, 26, 850, 28, 250, 842, 1300, 962, 33, 1220, 1450, 1300, 50, 1370, 1810, 1700, 234, 1682, 2500, 1850, 610, 260, 2650, 2210, 170, 50, 130, 2900, 850, 2810, 140
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
-
Mathematica
f[p_, e_] := If[e == 1, p^2 + 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
-
PARI
a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1 + f[i,1]^2, 1 + f[i,1]^f[i,2]));}
Formula
Multiplicative with a(p) = p^2 + 1 and a(p^e) = p^e + 1 for e >= 2.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(s-1) - 1/p^(2*s-3) + 1/p^(3*s-3) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^6 - 2/p^7 + 1/p^8) = 0.73644353930922037459... .