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

1,2

Links

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

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... .

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]

Prog

(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])); }

Crossrefs

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

Cf. A002117, A013661, A183699.

Sequence in context: A005093 A328639 A351265 * A365479 A374537 A360560

Adjacent sequences: A369715 A369716 A369717 * A369719 A369720 A369721

Keyword

nonn,easy,mult,look

Author

Amiram Eldar, Jan 30 2024

Status

approved