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A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

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%I A340088 #14 Dec 31 2020 08:21:01
%S A340088 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 A340088 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 A340088 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 A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.
%C A340088 Conjecture: a(n) = 1 iff n = 1 or in A050376. This is an infinitary analog of Lehmer's totient conjecture from 1932.
%C A340088 For all i, j > 1: a(i) = a(j) => A302777(i) = A302777(j), if the above conjecture holds.
%H A340088 Antti Karttunen, <a href="/A340088/b340088.txt">Table of n, a(n) for n = 1..65537</a>
%H A340088 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer&#39;s_totient_problem">Lehmer's totient problem</a>
%F A340088 a(n) = A091732(n) / A340087(n) = A091732(n) / gcd(n-1, A091732(n)).
%F A340088 For all n >= 1, a(A084400(n)) = 1.
%o A340088 (PARI)
%o A340088 ispow2(n) = (n && !bitand(n,n-1));
%o A340088 A302777(n) = ispow2(isprimepower(n));
%o A340088 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 A340088(n) = { my(x=A091732(n)); (x/gcd(n-1, x)); };
%Y A340088 Cf. A050376, A084400, A091732, A302777, A340087, A340089.
%Y A340088 Cf. also A160595.
%K A340088 nonn
%O A340088 1,6
%A A340088 _Antti Karttunen_, Dec 31 2020