%I #9 Jul 20 2022 18:43:34

%S 1,-1,-1,0,-1,-4,-1,0,0,0,-1,7,-1,0,-1,0,-1,8,-1,1,1,0,-1,-10,0,0,0,

%T -25,-1,10,-1,0,-1,0,1,15,-1,0,1,-8,-1,6,-1,-1,2,0,-1,16,0,2,-1,1,-1,

%U -6,1,46,1,0,-1,-9,-1,0,0,0,1,10,-1,1,-1,2,-1,-29,-1,0,4,-1,1,6,-1,16,0,0,-1,29,1,0,-1,2,-1,-8

%N Dirichlet inverse of A009194, the greatest common divisor of sigma(n) and n, where sigma is the sum of divisors function.

%H Antti Karttunen, <a href="/A355829/b355829.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A009194(n/d) * a(d).

%t s[n_] := GCD[n, DivisorSigma[1, n]]; a[1] = 1; a[n_] := - DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Jul 20 2022 *)

%o (PARI)

%o A009194(n) = gcd(n, sigma(n));

%o memoA355829 = Map();

%o A355829(n) = if(1==n,1,my(v); if(mapisdefined(memoA355829,n,&v), v, v = -sumdiv(n,d,if(d<n,A009194(n/d)*A355829(d),0)); mapput(memoA355829,n,v); (v)));

%Y Cf. A000203, A009194.

%Y Cf. also A355828.

%K sign

%O 1,6

%A _Antti Karttunen_, Jul 20 2022