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

Author

Amiram Eldar, Jan 30 2024

Keywords

• nonn
• easy
• mult
• look

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A034448, A001694, A197863, A369716, A369717, A369721.

Cf. A002117, A013661, A183699.

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

a(n) = A034448(A197863(n)).

Multiplicative with a(p) = p^2 + 1 and a(p^e) = p^e + 1 for e >= 2.

a(n) >= A034448(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^(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... .