A124385 - OEIS

%I #21 Feb 22 2023 10:20:03

%S 1,1,1,0,1,-1,1,1,1,-1,1,-1,1,1,1,1,1,-3,1,1,1,1,1,-3,-1,3,5,-1,1,-7,

%T 1,9,5,-1,5,-13,1,13,11,-5,1,-25,1,25,29,5,1,-53,19,19,53,23,1,-95,

%U -73,81,81,25,1,-119,1,119,27,1,113,-143,1,89,243,-69,1,-335,1,457,351,-81,145,-841,1,497,831,425,1,-1809,-883,1809

%N For all n >= 2, Sum_{2<=k<=n, gcd(k,n)>1} a(k) = 1. a(1)=1.

%H Antti Karttunen, <a href="/A124385/b124385.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = 1, and for n > 1, a(n) = 1 - Sum_{2 <= k <= n-1} [gcd(k,n)>1]*a(k), where [ ] is the Iverson bracket. - _Antti Karttunen_, Feb 22 2023

%e The positive integers which are <= 6 and are not coprime to 6 are 2,3,4,6. And a(6) is such that a(2)+a(3)+a(4)+a(6) = 1, i.e. a(6) = 1 - (a(2)+a(3)+a(4)) = 1 - 2 = -1.

%e The positive integers which are <= 12 and are not coprime to 12 2,3,4,6,8,9,10,12. And a(12) is such that a(2)+a(3)+a(4)+a(6)+a(8)+a(9)+a(10)+a(12) = 1.

%t f[n_] := Select[Range[2, n], GCD[ #, n] > 1 &];g[l_] :=Append[l, 1 - Plus @@ l[[Most[f[Length[l] + 1]]]]];Nest[g, {1}, 85] (* _Ray Chandler_, Nov 13 2006 *)

%o (PARI)

%o up_to_n = 10000;

%o A124385list(up_to_n) = { my(v=vector(up_to_n)); v[1] = 1; for(n=2,up_to_n,v[n] = 1-sum(k=2,n-1,(gcd(k,n)>1)*v[k])); (v); };

%o v124385 = A124385list(up_to_n);

%o A124385(n) = v124385[n]; \\ _Antti Karttunen_, Feb 22 2023

%Y Cf. A124386.

%K sign

%O 1,18

%A _Leroy Quet_, Oct 29 2006

%E Extended by _Ray Chandler_, Nov 13 2006