A350320 Totient numbers k such that 10*k is a nontotient.

110, 13310, 18260, 78980, 130460, 143660, 163460, 164780, 167420, 284900, 325160, 329780, 332420, 370700, 381260, 403700, 418220, 431420, 453860, 514580, 526460, 535700, 554180, 560780, 603020, 628100, 646580, 665060, 675620, 732380, 745580, 765380, 801020

Offset: 1

Author

Jianing Song, Dec 24 2021

Keywords

• nonn

Comments

10 is the smallest totient number that is not in A301587.

If 10*phi(m) is a nontotient, then m is divisible by 121 but not by 5, so every term is divisible by 110.

Proof. In the following cases, 10*phi(m) is a totient number:

(a) If m is not divisible by 11, then phi(11*m) = phi(11)*phi(m) = 10*phi(m).

(b) If m is divisible by 11 but not by 121 or 5, then phi((m/11)*125) = phi(m/11)*phi(125) = (phi(m)/10)*100 = 10*phi(m).

(c) If m is divisible by 5 but not by 2, then phi(4*5*m) = phi(4)*phi(5*m) = 2*(5*phi(m)) = 10*phi(m).

(d) If m is divisible by 5 and 2, then phi(10*m) = 10*phi(m).

So the only left case is that m is divisible by 121 but not by 5.

Examples

			110 is a term since 110 = phi(121) = phi(242), but phi(n) = 10*110 = 1100 has no solution.
13310 is a term since 13310 = phi(14641) = phi(29282), but phi(n) = 10*13310 = 133100 has no solution.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Totient numbers k such that m*k is a nontotient: A350316 (m=3), A350317 (m=5), A350318 (m=7), A350319 (m=9), this sequence (m=10), A350321 (m=14).

Cf. A002202, A005277, A301587.

Programs

PARI

isA350320(n) = istotient(n) && !istotient(10*n)