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

%I #33 Dec 29 2024 04:10:52

%S 399,2915,7055,63503,147455,1587599,1710863,2249999,2924099,6656399,

%T 9486399,14288399,19289663,25603599,26936099,28451555,31270463,

%U 32148899,45158399,49280399,71368703,91011599,105884099,111513599,144288143,146894399,150405695,152028899,175827599

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

%C Intersection of A005563 and A006972.

%C 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, ...

%C From _David A. Corneth_, Aug 26 2023: (Start)

%C 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.

%C 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)

%C 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

%H David A. Corneth, <a href="/A292538/b292538.txt">Table of n, a(n) for n = 1..2391</a> (terms <= 4*10^17; first 164 terms from Amiram Eldar)

%H Samuel S. Wagstaff, <a href="https://doi.org/10.1007/s11139-024-00846-1">Ramanujan's taxicab number and its ilk</a>, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; <a href="https://www.researchgate.net/publication/379960100_Ramanujan&#39;s_taxicab_number_and_its_ilk">ResearchGate link</a>, <a href="https://homes.cerias.purdue.edu/~ssw/taxi.pdf">author's copy</a>.

%p filter:= t ->

%p andmap(f -> f[2]=1 and (t+1) mod (f[1]+1) = 0, ifactors(t)[2]):

%p select(filter, [seq(k^2-1, k=3..10^5)]); # _Robert Israel_, Sep 24 2017

%t 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]

%Y Cf. A005563, A006972, A080262.

%K nonn

%O 1,1

%A _Amiram Eldar_, Sep 18 2017

%E More terms from _David A. Corneth_, Aug 26 2023