A292538 Lucas-Carmichael numbers of the form k^2 - 1.

399, 2915, 7055, 63503, 147455, 1587599, 1710863, 2249999, 2924099, 6656399, 9486399, 14288399, 19289663, 25603599, 26936099, 28451555, 31270463, 32148899, 45158399, 49280399, 71368703, 91011599, 105884099, 111513599, 144288143, 146894399, 150405695, 152028899, 175827599

Offset: 1

Author

Amiram Eldar, Sep 18 2017

Keywords

• nonn

Comments

Intersection of A005563 and A006972.

The numbers k such that k^2 - 1 is a Lucas-Carmichael number are 20, 54, 84, 252, 384, 1260, 1308, 1500, 1710, 2580, 3080, 3780, 4392, ...

From David A. Corneth, Aug 26 2023: (Start)

As k^2 - 1 = (k - 1)*(k + 1) and k is even we have k-1 and k+1 are coprime. So we can factor k-1 and k+1 separately when checking if k^2 - 1 is a term.

Possible other ideas are factoring an odd number only once, keeping it for the factorization of k^2 - 1 and (k + 2)^2 - 1. Alternatively dodging k = 18m +- 8, 18m +- 10 or 50m +- 24, 50m +- 26 to not get numbers that are multiples of odd primes squared. (End)

Wagstaff (2024) found that among the first 10^4 Lucas-Carmichael numbers there are 164 that are also Cunningham numbers (A080262) and that all of them are in this sequence. Below 10^15 there are 682 Lucas-Carmichael numbers that are also Cunningham numbers, and all of them are in this sequence (checked using the list of Lucas-Carmichael numbers by Daniel Suteu at A006972). - Amiram Eldar, Dec 29 2024

Links

• David A. Corneth, Table of n, a(n) for n = 1..2391 (terms <= 4*10^17; first 164 terms from Amiram Eldar)
• Samuel S. Wagstaff, Ramanujan's taxicab number and its ilk, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; ResearchGate link, author's copy.

Crossrefs

Cf. A005563, A006972, A080262.

Programs

Maple

filter:= t ->
  andmap(f -> f[2]=1 and (t+1) mod (f[1]+1) = 0, ifactors(t)[2]):
select(filter, [seq(k^2-1, k=3..10^5)]); # Robert Israel, Sep 24 2017

Mathematica

lcQ[n_] := !PrimeQ[n] && Union[Transpose[FactorInteger[n]][[2]]] == {1} && Union[Mod[n + 1, Transpose[FactorInteger[n]][[1]] + 1]] == {0}; Select[Range[2, 10^4]^2 - 1, lcQ]

Extensions

More terms from David A. Corneth, Aug 26 2023