A351464 -id:A351464 - cp's OEIS frontend

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

Rémy Sigrist, Feb 11 2022

			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.

Cf. A289311, A351464, A351475.

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

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 *)

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

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

Original entry on oeis.org

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

Rémy Sigrist, Feb 12 2022

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

			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.

Cf. A351464, A351465, A289320.

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

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 *)

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

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

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI