A316524 Signed sum over the prime indices of n.

0, 1, 2, 0, 3, -1, 4, 1, 0, -2, 5, 2, 6, -3, -1, 0, 7, 1, 8, 3, -2, -4, 9, -1, 0, -5, 2, 4, 10, 2, 11, 1, -3, -6, -1, 0, 12, -7, -4, -2, 13, 3, 14, 5, 3, -8, 15, 2, 0, 1, -5, 6, 16, -1, -2, -3, -6, -9, 17, -1, 18, -10, 4, 0, -3, 4, 19, 7, -7, 2, 20, 1, 21, -11, 2, 8, -1, 5, 22, 3, 0, -12, 23, -2, -4, -13, -8, -4, 24

Offset: 1

Author

Gus Wiseman, Jul 05 2018

Keywords

• sign

Comments

If n = prime(x_1) * prime(x_2) * prime(x_3) * ... * prime(x_k) then a(n) = x_1 - x_2 + x_3 - ... + (-1)^(k-1) x_k, where the x_i are weakly increasing positive integers.

The value of a(n) depends only on the squarefree part of n, A007913(n). - Antti Karttunen, May 06 2022

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000040, A000607, A007913, A071321, A100118, A175352, A316313, A316523.

Cf. A027746, A112798, A119899 (positions of negative terms).

Cf. A344616 (absolute values), A344617 (signs).

Programs

Mathematica

Table[Sum[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]][[k]]*(-1)^(k-1),{k,PrimeOmega[n]}],{n,100}]

PARI

a(n) = {my(f = factor(n), vp = []); for (k=1, #f~, for( j=1, f[k,2], vp = concat (vp, primepi(f[k,1])));); sum(k=1, #vp, vp[k]*(-1)^(k+1));} \\ Michel Marcus, Jul 06 2018

Python

from sympy import factorint, primepi
def A316524(n):
    fs = [primepi(p) for p in factorint(n,multiple=True)]
    return sum(fs[::2])-sum(fs[1::2]) # Chai Wah Wu, Aug 23 2021

Formula

a(n) = A344616(n) * A344617(n) = a(A007913(n)). - Antti Karttunen, May 06 2022

Extensions

More terms from Antti Karttunen, May 06 2022