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A350085
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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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3
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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, 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, 2, 10, 2, 22, 6, 2, 2, 18, 2, 2, 6, 6, 46, 2, 2, 2, 6, 2, 2, 2, 2, 10, 2
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OFFSET
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1,1
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COMMENTS
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Conjecture: a(n) != 0 for all n.
Records: 30 (A007617(n) = 3), 54 (A007617(n) = 21), 106 (A007617(n) = 90), 2010 (A007617(n) = 450), ...
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).
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.
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LINKS
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Michel Marcus, Table of n, a(n) for n = 1..7626
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EXAMPLE
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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.
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.
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.
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.
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PROG
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(PARI) b(n) = if(!istotient(n), for(k=2, oo, if(istotient(k) && !istotient(n*k), return(k))))
list(lim) = my(v=[]); for(n=1, lim, if(!istotient(n), v=concat(v, b(n)))); v \\ gives a(n) for A007617(n) <= lim
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CROSSREFS
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Cf. A007617, A350086, A301587.
Sequence in context: A277982 A287921 A073401 * A040875 A131773 A091746
Adjacent sequences: A350082 A350083 A350084 * A350086 A350087 A350088
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KEYWORD
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nonn,changed
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AUTHOR
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Jianing Song, Dec 12 2021
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STATUS
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approved
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