A351475 Multiplicative with a(prime(k)^e) = k^2 + e^2 for any k, e > 0.

1, 2, 5, 5, 10, 10, 17, 10, 8, 20, 26, 25, 37, 34, 50, 17, 50, 16, 65, 50, 85, 52, 82, 50, 13, 74, 13, 85, 101, 100, 122, 26, 130, 100, 170, 40, 145, 130, 185, 100, 170, 170, 197, 130, 80, 164, 226, 85, 20, 26, 250, 185, 257, 26, 260, 170, 325, 202, 290, 250

Offset: 1

Author

Rémy Sigrist, Feb 12 2022

Keywords

• nonn
• mult

Comments

This sequence gives the norm of the function f defined in A351464-A351465.

Examples

			For n = 42:
- 42 = 2 * 3 * 7 = prime(1)^1 * prime(2)^1 * prime(4)^1,
- a(42) = (1^2 + 1^2) * (2^2 + 1^2) * (4^2 + 1^2) = 170.
		

Crossrefs

Cf. A351464, A351465, A289320.

Programs

Maple

a:= proc(n) option remember; uses numtheory;
      mul(pi(i[1])^2+i[2]^2, i=ifactors(n)[2])
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 15 2022

Mathematica

f[p_, e_] := PrimePi[p]^2 + e^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 15 2022 *)

PARI

a(n) = { my (f=factor(n), p=f[, 1]~, e=f[, 2]~); prod (k=1, #p, primepi(p[k])^2 + e[k]^2) }

Formula

a(n) = A351464(n)^2 + A351465(n)^2.