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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.
3
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, 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, 2, 10, 46, 2, 2, 2, 2, 2, 6, 78, 2, 10, 2, 10, 46, 10, 2, 46, 2
OFFSET
1,1
COMMENTS
Subsequence of A350085.
Conjecture: a(n) != 0 for all n.
Records: 22 (A005277(n) = 14), 106 (A005277(n) = 90), 2010 (A005277(n) = 450), ...
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).
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.
LINKS
EXAMPLE
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.
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.
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.
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.
PROG
(PARI) b(n) = if(!istotient(n), for(k=2, oo, if(istotient(k) && !istotient(n*k), return(k))))
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Dec 12 2021
STATUS
approved