A351434 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)^(k_j + 1)).

1, 1, 4, 1, 16, 4, 36, 1, 8, 16, 100, 4, 144, 36, 64, 1, 256, 8, 324, 16, 144, 100, 484, 4, 64, 144, 16, 36, 784, 64, 900, 1, 400, 256, 576, 8, 1296, 324, 576, 16, 1600, 144, 1764, 100, 128, 484, 2116, 4, 216, 64, 1024, 144, 2704, 16, 1600, 36, 1296, 784, 3364, 64, 3600, 900, 288, 1, 2304

Offset: 1

Author

Ilya Gutkovskiy, Feb 11 2022

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A003557, A003958, A003959, A023900, A064549, A326297, A327564, A351419, A351435.

Programs

Maple

a:= n-> mul((i[1]-1)^(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..65);  # 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, 65}]

PARI

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

Formula

a(n) = A003958(n) * |A023900(n)|.

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (1 - (3*p^2 - 4*p + 2)/(p*(p^3 - p + 1))) = 0.1161464566... . - Amiram Eldar, Nov 19 2022