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

1,4

Links

Table of n, a(n) for n=1..80.

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A289311, A351464, A351475.

Sequence in context: A290089 A323902 A331991 * A309634 A309633 A329632

Adjacent sequences: A351462 A351463 A351464 * A351466 A351467 A351468

Keyword

nonn,easy

Author

Rémy Sigrist, Feb 11 2022

Status

approved