Voozh

A272295

a(n) = v(2*n+1), where v is a completely multiplicative sequence defined by v(2)=0 and for prime p >= 3: v(p^e) = ((p+1)/2)^e when p == 1 mod 4, and v(p^e) = ((1-p)/2)^e when p == 3 mod 4.

1, -1, 3, -3, 1, -5, 7, -3, 9, -9, 3, -11, 9, -1, 15, -15, 5, -9, 19, -7, 21, -21, 3, -23, 9, -9, 27, -15, 9, -29, 31, -3, 21, -33, 11, -35, 37, -9, 15, -39, 1, -41, 27, -15, 45, -21, 15, -27, 49, -5, 51, -51, 9, -53, 55, -19, 57, -33, 7, -27, 25, -21, 27, -63

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OFFSET

0,3

COMMENTS

If n = prime(k) > 2, a(n) = A271974(k).

LINKS

Aloe Poliszuk, Table of n, a(n) for n = 0..10000

Dimitris Valianatos, Comments on this sequence, Apr 25 2016

FORMULA

Sum_{n >= 1, n not divisible by 2 or 3} 1/v(n) = 1.

Conjecture: Sum_{n >= 1, n not divisible by 2 or 3} (moebius(n)/v(n))^2 = 7/5 = 1.4.

EXAMPLE

v[0..10] = 1, 1, 0, -1, 0, 3, 0, -3, 0, 1, 0.

v(35) = v(5) * v(7) = 3 * (-3) = -9.

MAPLE

v := p -> ifelse(modp(p, 4)<>1, iquo(1-p, 2), iquo(1+p, 2)):

a := n -> mul(v(pe[1])^pe[2], pe in ifactors(2*n+1)[2]):

seq(a(n), n = 0..59); # Peter Luschny, Jun 05 2025

MATHEMATICA

f[p_, e_] := Quotient[If[Mod[p, 4]!=1, 1-p, 1+p], 2]^e; a[0] = 1; a[n_] := Times @@ f @@@ FactorInteger[2*n+1]; Array[a, 59, 0] (* Peter Luschny, Jun 05 2025 *)

PROG

(PARI) a(n)=fa=factorint(2*n+1); dv=fa[, 1]; pl=#dv; ml=fa[, 2]; g=1; for(i=1, pl, ds=dv[i]; v=1; if(ds%4==1, v*=(1+ds)\2, v*=(1-ds)\2); for(k=1, ml[i], g*=v)); return(g);

(PARI) {

forstep(n=1, 120, 2,

fa=factorint(n); dv=fa[, 1]; pl=#dv; ml=fa[, 2];

g=1;

for(i=1, pl,

ds=dv[i]; v=1;

if(ds%4==1, v*=(1+ds)\2, v*=(1-ds)\2);

for(k=1, ml[i], g*=v)

);

print1(g", ")

);

}

(PARI) a(n) = {my(f = factor(2*n+1)); for (k=1, #f~, if (f[k, 1] == 2, f[k, 1] = 0, if (f[k, 1] % 4 == 1, f[k, 1] = (1+f[k, 1])/2, f[k, 1] = (1-f[k, 1])/2)); ); factorback(f); } \\ Michel Marcus, May 02 2016

(Python)

from sympy import factorint

from math import prod

def a(n) -> int:

v = lambda p: (1-p)//2 if p&2 else (1+p)//2

h = lambda n: prod(v(p)**e for p, e in factorint(n).items())

return h(2*n+1)

print([a(n) for n in range(60)]) # David Radcliffe, Jun 04 2025

CROSSREFS

Cf. A271974.

Sequence in context: A318319 A321785 A021755 * A164984 A248810 A339904

Adjacent sequences: A272292 A272293 A272294 * A272296 A272297 A272298