A344767 Möbius transform of A011772.

1, 2, 1, 4, 3, -1, 5, 8, 6, -2, 9, 1, 11, -1, 0, 16, 15, -1, 17, 7, -1, -1, 21, -1, 20, -2, 18, -4, 27, 11, 29, 32, 0, -2, 5, -5, 35, -1, -1, -8, 39, 14, 41, 17, -2, -1, 45, 1, 42, 0, 0, 23, 51, 1, -3, 33, -1, -2, 57, -12, 59, -1, 15, 64, 10, 0, 65, -4, 0, 7, 69, 48, 71, -2, -1, 33, 6, 1, 77, 33, 54, -2, 81, 27, 15, -1, 0

Offset: 1

Author

Antti Karttunen, May 30 2021

Keywords

• sign
• look

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A008683, A011772, A344768.

Programs

PARI

A344767(n) = sumdiv(n,d,moebius(n/d)*A011772(d));

Python

from itertools import combinations
from math import prod
from sympy import factorint, divisors
from sympy.ntheory.modular import crt
from sympy.ntheory import mobius
def A011772(n):
    plist = [p**q for p, q in factorint(2*n).items()]
    return 2*n-1 if len(plist) == 1 else 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)))
def A344767(n): return sum(mobius(n//d)*A011772(d) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 20 2021

Formula

a(n) = Sum_{d|n} A008683(n/d) * A011772(d).