A385542 The sum of the aliquot divisors of n that are powerful.
0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 13, 1, 10, 1, 5, 1, 1, 1, 13, 1, 1, 10, 5, 1, 1, 1, 29, 1, 1, 1, 14, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 29, 1, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 61, 1, 1, 1, 5, 1, 1, 1, 58, 1, 1, 26, 5, 1, 1, 1, 29, 37
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
- Index entries for sequences related to powerful numbers.
Programs
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Mathematica
f[p_, e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 0; a[n_] := Times @@ f @@@ (fct = FactorInteger[n]) - If[AllTrue[fct[[;;, 2]], # > 1 &], n, 0]; Array[a, 100]
-
PARI
a(n) = {my(f = factor(n), s); s = prod(i=1, #f~, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1) - f[i,1]); if(n==1 || vecmin(f[,2]) > 1, s -= n); s};
Formula
a(n) = Sum_{d|n, d < n} A112526(d) * d.
a(n) = 1 if and only if n is either a squarefree number (A005117) > 1 or a square of a prime (A001248), i.e., if and only if n is in A167207 \ {1}.
Dirichlet g.f.: (zeta(s) - 1)* zeta(2*s-2) * zeta(3*s-3) / zeta(6*s-6).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)*(zeta(3/2)-1)/(3*zeta(3)) = 1.168033893310319119603... .
More precise asymptotics: Sum_{k=1..n} a(k) ~ (zeta(3/2) - 1)*zeta(3/2)*n^(3/2) / (3*zeta(3)) + 3*zeta(2/3)*(zeta(4/3) - 1)*n^(4/3) / (2*Pi^2) - n/2. - Vaclav Kotesovec, Jul 03 2025