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

1,3

Comments

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

Links

Table of n, a(n) for n=1..78.

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) = 1.

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A289310, A351465, A351475.

Sequence in context: A367583 A302788 A057499 * A363943 A241919 A286469

Adjacent sequences: A351461 A351462 A351463 * A351465 A351466 A351467

Keyword

sign,easy

Author

Rémy Sigrist, Feb 11 2022

Status

approved