A350085 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.

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

Offset: 1

Author

Jianing Song, Dec 12 2021

Keywords

• nonn

Comments

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.

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

Examples

			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.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..7626

Crossrefs

Cf. A007617, A350086, A301587.

Programs

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