A277389 Numbers k such that lambda(k)^3 divides (k-1)^2, where lambda(k) = A002322(k).

1, 2, 1729, 19683001, 367804801, 631071001, 2064236401, 2320690177, 24899816449, 40017045601, 110592000001, 137299665601, 432081216001, 479534887801, 760355883001, 1111195454401, 3176523000001, 3495866888449, 3837165696001, 8571867768001, 14373832968001

Offset: 1

Author

Thomas Ordowski, Oct 12 2016

Keywords

• nonn

Comments

Carmichael numbers are composite numbers n such that k = 1 (mod lambda(k)); equivalently, lambda(k)^2 divides (k-1)^2. As a result, all composite terms of the sequence are Carmichael numbers A002997. But there are no primes in this sequence except for 2 (since lambda(p) = p-1 and (p-1)^3 > (p-1)^2 for p > 2) and so all terms in this sequence other than 1 and 2 are Carmichael numbers. - Charles R Greathouse IV, Oct 15 2016

Is this sequence infinite?

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1469 (terms below 10^22, calculated using data from Claude Goutier; terms 1..58 from Robert Israel, terms 59..101 from Charles R Greathouse IV)
• Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Crossrefs

Cf. A002322, A002997, A265628.

Programs

PARI

isok(n) = ((n-1)^2 % (lcm(znstar(n)[2])^3)) == 0; \\ Michel Marcus, Oct 12 2016

Extensions

a(4) from Michel Marcus, Oct 12 2016

a(5)-a(6) from Michel Marcus, Oct 13 2016

More terms from Robert Israel, Oct 13 2016