A351465 -id:A351465 - cp's OEIS frontend

A351464 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 real part of f(n). See A351465 for the imaginary part.

1, 1, 2, 1, 3, 1, 4, 1, 2, 2, 5, 0, 6, 3, 5, 1, 7, 0, 8, 1, 7, 4, 9, -1, 3, 5, 2, 2, 10, 0, 11, 1, 9, 6, 11, -2, 12, 7, 11, 0, 13, 1, 14, 3, 4, 8, 15, -2, 4, 1, 13, 4, 16, -1, 14, 1, 15, 9, 17, -5, 18, 10, 6, 1, 17, 2, 19, 5, 17, 4, 20, -4, 21, 11, 4, 6, 19, 3

Offset: 1

Rémy Sigrist, Feb 11 2022

Apparently, each integer (from Z) appears in this sequence.

			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) = 1.

Cf. A289310, A351465, A351475.

b:= proc(n) option remember; uses numtheory;
      mul(pi(i[1])+i[2]*I, i=ifactors(n)[2])
    end:
a:= n-> Re(b(n)):
seq(a(n), n=1..78);  # Alois P. Heinz, Feb 15 2022

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

a(n) = { my (f=factor(n), p=f[,1]~, e=f[,2]~); real(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

A351464 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 real part of f(n). See A351465 for the imaginary part.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI