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a(n) is the smallest totient number k > 1 such that A007617(n)*k is a nontotient number, or 0 if no such number exists.
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%I #26 Jun 25 2023 08:11:19

%S 30,10,2,10,22,2,22,6,2,2,54,10,2,22,22,6,2,18,2,10,2,2,6,6,2,2,2,2,

%T 22,10,6,10,2,2,2,2,18,6,2,10,6,2,2,10,6,2,2,2,30,10,2,6,2,6,106,2,2,

%U 2,10,2,22,6,2,2,18,2,2,6,6,46,2,2,2,6,2,2,2,2,10,2

%N a(n) is the smallest totient number k > 1 such that A007617(n)*k is a nontotient number, or 0 if no such number exists.

%C Conjecture: a(n) != 0 for all n.

%C Records: 30 (A007617(n) = 3), 54 (A007617(n) = 21), 106 (A007617(n) = 90), 2010 (A007617(n) = 450), ...

%C By definition, a totient number N > 1 is a term if and only if there exists a 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 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 A005277(n) = 34, 86, 68, 186, 14, 902, 318.

%C Most of the terms are of the form prime-1, but there are some exceptions. Here is a list of distinct such instances [not(prime-1), the corresponding least nontotient]: [54, 21], [110, 10450], [342, 29214], [506, 63250], [3422, 100050], [294, 118062], [2162, 235824], [1210, 308660], [930, 395070], ... That is, instead of k being invphi(prime), in these cases it appears that k is invphi(prime^q). - _Michel Marcus_, Jun 25 2023

%H Michel Marcus, <a href="/A350085/b350085.txt">Table of n, a(n) for n = 1..7626</a>

%e A007617(55) = 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(55) = 106.

%e A007617(307) = 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(307) = 2010.

%e A007617(637) = 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(637) = 28.

%e A007617(194495) = 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(194495) = 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 (PARI) b(n) = if(!istotient(n), for(k=2, oo, if(istotient(k) && !istotient(n*k), return(k))))

%o list(lim) = my(v=[]); for(n=1, lim, if(!istotient(n), v=concat(v,b(n)))); v \\ gives a(n) for A007617(n) <= lim

%Y Cf. A007617, A350086, A301587.

%K nonn

%O 1,1

%A _Jianing Song_, Dec 12 2021