A354988 a(n) = A345993(n) - A345992(n).

0, 1, 2, 3, 4, 1, 6, 7, 8, 3, 10, 1, 12, 5, -2, 15, 16, 7, 18, 1, 4, 9, 22, -5, 24, 11, 26, -3, 28, 1, 30, 31, -8, 15, -2, 5, 36, 17, 10, 3, 40, 1, 42, -7, -4, 21, 46, 13, 48, 23, -14, 9, 52, 25, 6, 1, 16, 27, 58, -11, 60, 29, -2, 63, 8, -5, 66, 13, -20, -9, 70, 1, 72, 35, 22, -15, 4, 7, 78, 11, 80, 39, 82, 17, -12

Offset: 1

Author

Antti Karttunen, Jun 16 2022

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Crossrefs

Cf. A345992, A345993, A345995 (positions of negative terms), A354989 (their characteristic function).

Absolute values differ from A076388 for the first time at n=60, where a(60) = -11, while A076388(60) = 7.

Programs

Mathematica

a[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), n], m++]; GCD[n, m + 1] - GCD[n, m]]; Array[a, 100] (* Amiram Eldar, Jun 16 2022 *)

PARI

A354988(n) = for(m=1, oo, if((m*(m+1))%n==0, return(gcd(n,1+m)-gcd(n,m))));
(Python 3.8+)
from math import gcd, prod
from itertools import combinations
from sympy import factorint
from sympy.ntheory.modular import crt
def A354988(n):
    if n == 1:
        return 0
    plist = tuple(p**q for p, q in factorint(n).items())
    return n-1 if len(plist) == 1 else -gcd(n,s:=int(min(min(crt((m, n//m), (0, -1))[0], crt((n//m, m), (0, -1))[0]) for m in (prod(d) for l in range(1, len(plist)//2+1) for d in combinations(plist, l))))) + gcd(n,s+1) # Chai Wah Wu, Jun 16 2022