A063668 Numbers of the form 12*k + 2 with nonempty inverse totient set.

2, 110, 506, 2162, 3422, 4970, 6806, 11342, 13310, 17030, 27722, 31862, 36290, 51302, 56882, 62750, 68906, 96410, 120062, 128522, 146306, 175142, 185330, 195806, 217622, 228962, 240590, 252506, 267674, 316406, 343982, 358202, 417962, 433622, 465806, 516242

Offset: 1

Author

Labos Elemer, Aug 22 2001

Keywords

• nonn

Comments

Except for the first term, each of these sets contains 2 terms. Other numbers of the form 12*k + 2 have empty inverse totient sets.

From Jianing Song, Dec 30 2018: (Start)

Except for the first term, these are numbers of the form (p - 1)*p^(2*e-1) = phi(p^(2*e)) where p is a prime congruent to 11 modulo 12. The inverse totient set for a(n) (n > 1) is {p^(2*e), 2*p^(2*e)}.

Numbers k such that A063667((k-2)/12) != 0.

The number of terms <= N is roughly (1/8)*sqrt(N)/log(N). (End)

Examples

			1407782 = 1186*1187 where 1187 is a prime congruent to 11 modulo 12, so 1407782 is a term, with invphi(1407782) = {1408969, 2817938} = {1187^2, 2*1187^2}.
267674 = 22*23^3 where 23 is a prime congruent to 11 modulo 12, so 267674 is a term, with invphi(267674) = {279841, 559682} = {23^4, 2*23^4}. - _Jianing Song_, Dec 30 2018
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..315 from Jianing Song)

Crossrefs

Cf. A000010, A063667.

Programs

PARI

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1)
isok(n) = my(p=A006530(n), e=if(n>1, valuation(n,p), 1)); (n==2) || (p%12==11&&e%2&&n==(p-1)*p^e) \\ Jianing Song, Dec 30 2018

PARI

isok(n) = #invphi(n) && !((n-2) % 12); \\ Michel Marcus, Dec 30 2018; using the invphi script by Max Alekseyev

PARI

isok(m) = !((m-2) % 12) && istotient(m); \\ Michel Marcus, Apr 20 2023

Extensions

Three missing terms added by Jianing Song, Dec 30 2018