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A350086 a(n) is the smallest totient number k > 1 such that A005277(n)*k is a nontotient number, or 0 if no such number exists.

%I A350086 #15 Mar 11 2023 10:11:30
%S A350086 22,22,2,2,22,2,10,10,2,6,106,2,22,46,2,2,2,6,2,10,2,2,6,2,78,2,18,2,
%T A350086 6,2,2,2,2,46,58,2,2,2,58,2,6,2,2,2,10,10,2,46,2,2,2,82,2,30,2,6,2,10,
%U A350086 2,10,46,2,2,2,2,2,6,78,2,10,2,10,46,10,2,46,2
%N A350086 a(n) is the smallest totient number k > 1 such that A005277(n)*k is a nontotient number, or 0 if no such number exists.
%C A350086 Subsequence of A350085.
%C A350086 Conjecture: a(n) != 0 for all n.
%C A350086 Records: 22 (A005277(n) = 14), 106 (A005277(n) = 90), 2010 (A005277(n) = 450), ...
%C A350086 By definition, a totient number N > 1 is a term if and only if there exists an even nontotient r such that: (i) k*r is a totient for totient numbers 2 <= k < N; (ii) N*r is a nontotient. No term can be of the form m*m', where m > 1 is a totient and m' > 1 is in A301587 (otherwise m*r is a totient implies m*m'*r is a totient).
%C A350086 Conjecture: every totient number > 1 which is not of the form m*m', where m > 1 is a totient and m' > 1 is in A301587, appears in this sequence. For example, the numbers 2, 6, 10, 18, 22, 28, 30 first appears when A007617(n) = 7, 15, 5, 33, 11, 902, 3.
%H A350086 Michel Marcus, <a href="/A350086/b350086.txt">Table of n, a(n) for n = 1..8458</a>
%e A350086 A005277(11) = 90. N = 106 is a totient number > 1 such that 90*k is a totient for totient numbers 2 <= k < N, and 90*N is a nontotient, so a(11) = 106.
%e A350086 A005277(83) = 450. N = 2010 is a totient number > 1 such that 450*k is a totient for totient numbers 2 <= k < N, and 450*N is a nontotient, so a(83) = 2010.
%e A350086 A005277(187) = 902. N = 28 is a totient number > 1 such that 902*k is a totient for totient numbers 2 <= k < N, and 902*N is a nontotient, so a(187) = 28.
%e A350086 A005277(73991) = 241010. N = 100 is a totient number > 1 such that 241010*k is a totient for totient numbers 2 <= k < N, and 241010*N is a nontotient, so a(73991) = 100. Note that although 100 = 10*10 is a product of 2 totient number > 1, neither factor is in A301587, so nothing prevents that 100 is a term of this sequence.
%o A350086 (PARI) b(n) = if(!istotient(n), for(k=2, oo, if(istotient(k) && !istotient(n*k), return(k))))
%o A350086 list(lim) = my(v=[]); forstep(n=2, lim, 2, if(!istotient(n), v=concat(v,b(n)))); v \\ gives a(n) for A005277(n) <= lim
%Y A350086 Cf. A005277, A350085, A301587.
%K A350086 nonn
%O A350086 1,1
%A A350086 _Jianing Song_, Dec 12 2021