A340088 - OEIS

%I #14 Dec 31 2020 08:21:01

%S 1,1,1,1,1,2,1,3,1,4,1,6,1,6,4,1,1,8,1,12,3,10,1,6,1,12,8,2,1,8,1,15,

%T 5,16,12,24,1,18,12,4,1,12,1,30,8,22,1,30,1,24,16,12,1,16,20,18,9,28,

%U 1,24,1,30,24,5,3,4,1,48,11,8,1,24,1,36,24,18,15,24,1,60,1,40,1,36,16,42,28,10,1,32,4,66

%N a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

%C Conjecture: a(n) = 1 iff n = 1 or in A050376. This is an infinitary analog of Lehmer's totient conjecture from 1932.

%C For all i, j > 1: a(i) = a(j) => A302777(i) = A302777(j), if the above conjecture holds.

%H Antti Karttunen, <a href="/A340088/b340088.txt">Table of n, a(n) for n = 1..65537</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer&#39;s_totient_problem">Lehmer's totient problem</a>

%F a(n) = A091732(n) / A340087(n) = A091732(n) / gcd(n-1, A091732(n)).

%F For all n >= 1, a(A084400(n)) = 1.

%o (PARI)

%o ispow2(n) = (n && !bitand(n,n-1));

%o A302777(n) = ispow2(isprimepower(n));

%o A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((d<n)&&A302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };

%o A340088(n) = { my(x=A091732(n)); (x/gcd(n-1, x)); };

%Y Cf. A050376, A084400, A091732, A302777, A340087, A340089.

%Y Cf. also A160595.

%K nonn

%O 1,6

%A _Antti Karttunen_, Dec 31 2020