A283656 Numbers n such that gcd(phi(n), n-1) > lambda(n).

65, 91, 217, 273, 451, 481, 703, 793, 1281, 1729, 1891, 1921, 2465, 2701, 3201, 4033, 4097, 4681, 5833, 6643, 6697, 7105, 7161, 8321, 8401, 8911, 9073, 10649, 11041, 11476, 11521, 12403, 12545, 13051, 14689, 14701, 15841, 16385, 16401, 16471, 18361, 18705, 18721, 19684, 19951, 20801, 21953, 22177, 22681, 23001

Offset: 1

Author

Thomas Ordowski and Altug Alkan, Mar 23 2017

Keywords

• nonn

Comments

All terms are composite. No powers of primes.

Contains all Carmichael numbers except A264012.

If n is in the sequence, then n-1 is not squarefree.

Problem: are there infinitely many such even numbers? : 11476, 19684, 24564, 37576, 57226, 65026, 80476, 89776, 91356, ...

It is possible to show there are infinitely many Carmichael numbers with the property. In fact this follows with a small modification of the original proof of the infinitude of the Carmichael numbers. It seems harder though to prove that there are infinitely many non-Carmichaels with the property, though undoubtedly it's true. - Carl Pomerance, Mar 24 2017

Links

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

Crossrefs

Cf. A000010, A002322, A002997, A049559, A264012.

Programs

Mathematica

Select[Range[10^4], GCD[EulerPhi[#], #-1] > CarmichaelLambda[#] &] (* Amiram Eldar, Aug 26 2019 *)