A351475 -id:A351475 - 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])) }

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.

Original entry on oeis.org

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])) }

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.

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