A351465 Let f be multiplicative with f(prime(k)^e) = k + e*i for any k, e > 0 (where i denotes the imaginary unit); a(n) is the imaginary part of f(n). See A351464 for the real part.

0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 5, 4, 1, 4, 1, 7, 6, 6, 1, 7, 2, 7, 3, 9, 1, 10, 1, 5, 7, 8, 7, 6, 1, 9, 8, 10, 1, 13, 1, 11, 8, 10, 1, 9, 2, 5, 9, 13, 1, 5, 8, 13, 10, 11, 1, 15, 1, 12, 10, 6, 9, 16, 1, 15, 11, 18, 1, 8, 1, 13, 7, 17, 9, 19, 1, 13

Offset: 1

Author

Rémy Sigrist, Feb 11 2022

Keywords

• nonn
• easy

Examples

			For n = 42:
- 42 = 2 * 3 * 7 = prime(1)^1 * prime(2)^1 * prime(4)^1,
- f(42) = (1+i) * (2+i) * (4+i) = 1 + 13*i,
- and a(42) = 13.
		

Crossrefs

Cf. A289311, A351464, A351475.

Programs

Maple

b:= proc(n) option remember; uses numtheory;
      mul(pi(i[1])+i[2]*I, i=ifactors(n)[2])
    end:
a:= n-> Im(b(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 15 2022

Mathematica

f[p_, e_] := PrimePi[p] + e*I; a[1] = 0; a[n_] := Im[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]~); imag(prod (k=1, #p, primepi(p[k]) + I*e[k])) }