A344763 a(n) = n - A011772(n).

0, -1, 1, -3, 1, 3, 1, -7, 1, 6, 1, 4, 1, 7, 10, -15, 1, 10, 1, 5, 15, 11, 1, 9, 1, 14, 1, 21, 1, 15, 1, -31, 22, 18, 21, 28, 1, 19, 27, 25, 1, 22, 1, 12, 36, 23, 1, 16, 1, 26, 34, 13, 1, 27, 45, 8, 39, 30, 1, 45, 1, 31, 36, -63, 40, 55, 1, 52, 46, 50, 1, 9, 1, 38, 51, 20, 56, 66, 1, 16, 1, 42, 1, 36, 51, 43, 58, 56, 1, 55

Offset: 1

Author

Antti Karttunen, May 30 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16383
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A001065, A011772, A344764, A344765, A344769.

Programs

Mathematica

A011772[n_] := Module[{m = 1}, While[!IntegerQ[(m(m+1))/(2n)], m++]; m];
a[n_] := n - A011772[n];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)

PARI

A344763(n) = (n-A011772(n));

Python

from sympy.ntheory.modular import crt
from sympy import factorint
from math import prod
from itertools import combinations
def A344763(n):
    plist = tuple(p**q for p, q in factorint(2*n).items())
    return 1-n if len(plist) == 1 else n-int(min(min(crt((m,2*n//m),(0,-1))[0],crt((2*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)))) # Chai Wah Wu, Jun 15 2022

Formula

a(n) = n - A011772(n).

a(n) = A344765(n) - A001065(n).

a(2^k) = 1-2^k. - Chai Wah Wu, Jun 15 2022