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A344767 Möbius transform of A011772.

%I A344767 #17 Jan 26 2025 20:28:25
%S A344767 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,
%T A344767 -4,27,11,29,32,0,-2,5,-5,35,-1,-1,-8,39,14,41,17,-2,-1,45,1,42,0,0,
%U A344767 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
%N A344767 Möbius transform of A011772.
%H A344767 Antti Karttunen, <a href="/A344767/b344767.txt">Table of n, a(n) for n = 1..16384</a>
%F A344767 a(n) = Sum_{d|n} A008683(n/d) * A011772(d).
%o A344767 (PARI) A344767(n) = sumdiv(n,d,moebius(n/d)*A011772(d));
%o A344767 (Python)
%o A344767 from itertools import combinations
%o A344767 from math import prod
%o A344767 from sympy import factorint, divisors
%o A344767 from sympy.ntheory.modular import crt
%o A344767 from sympy.ntheory import mobius
%o A344767 def A011772(n):
%o A344767     plist = [p**q for p, q in factorint(2*n).items()]
%o A344767     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)))
%o A344767 def A344767(n): return sum(mobius(n//d)*A011772(d) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jun 20 2021
%Y A344767 Cf. A008683, A011772, A344768.
%K A344767 sign,look
%O A344767 1,2
%A A344767 _Antti Karttunen_, May 30 2021