A351435 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j + 1)).

1, 9, 16, 27, 36, 144, 64, 81, 64, 324, 144, 432, 196, 576, 576, 243, 324, 576, 400, 972, 1024, 1296, 576, 1296, 216, 1764, 256, 1728, 900, 5184, 1024, 729, 2304, 2916, 2304, 1728, 1444, 3600, 3136, 2916, 1764, 9216, 1936, 3888, 2304, 5184, 2304, 3888, 512, 1944, 5184, 5292, 2916

Offset

1,2

Links

Table of n, a(n) for n=1..53.

Index entries for sequences computed from exponents in factorization of n.

Index entries for sequences computed from indices in prime factorization.

Formula

a(n) = A003959(n) * A048250(n).

Sum_{k=1..n} a(k) ~ c * n^3, where c = 1/(3 * Product_{p prime} (1 - p/(p^3-1))) = 1 /(3 * A065478) = 0.5787439255... . - Amiram Eldar, Nov 19 2022

Sum_{n>=1} 1/a(n) = zeta(2)/zeta(3) (A306633). - Amiram Eldar, Dec 15 2023

Maple

a:= n-> mul((i[1]+1)^(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..53);  # Alois P. Heinz, Feb 11 2022

Mathematica

f[p_, e_] := (p + 1)^(e + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Table[a[n], {n, 1, 53}]

Prog

(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 1]++; f[k, 2]++); factorback(f); \\ Michel Marcus, Feb 11 2022

Crossrefs

Cf. A003557, A003958, A003959, A048250, A064549, A065478, A306633, A326297, A327564, A351434.

Sequence in context: A183345 A303873 A257496 * A369642 A162616 A153362

Adjacent sequences: A351432 A351433 A351434 * A351436 A351437 A351438

Keyword

nonn,mult,easy

Author

Ilya Gutkovskiy, Feb 11 2022

Status

approved