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

1,3

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

Formula

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

Mathematica

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

Prog

(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

Crossrefs

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

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

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

Sequence in context: A277707 A357634 A344616 * A357630 A194549 A063277

Adjacent sequences: A316521 A316522 A316523 * A316525 A316526 A316527

Keyword

sign

Author

Gus Wiseman, Jul 05 2018

Extensions

More terms from Antti Karttunen, May 06 2022

Status

approved