%I #18 Apr 22 2024 13:44:00

%S 561,8911,10585,41041,115921,314821,334153,6313681,8134561,14913991,

%T 32914441,60957361,67902031,135556345,289766701,321197185,329769721,

%U 368113411,471905281,765245881,842202361,962442001,1507746241,2489462641,2588653081,3104207821

%N Triangular Carmichael numbers.

%C Intersection of A000217 and A002997.

%C The least triangular Carmichael numbers with the number of prime factors = 3, 4, 5, 6, 7, ... are: 561, 41041, 765245881, 321197185, 1583892181303201, ...

%C The number of terms below 10^k for k = 3, 4, ... are: 1, 2, 4, 7, 9, 13, 22, 32, 53, 77, 137, 211, 358, 545, 879, 1423, ...

%C Jonathan Vos Post discovered in 2004 that a(21) = 842202361 = A000217(41041) = A002817(286) is also a doubly triangular Carmichael number. The next number with this property is a(1108) = 292800629576356021 = A000217(765245881) = A002817(39121) (41041 and 765245881 are triangular Carmichael numbers that are also indices of triangular numbers that are also Carmichael numbers).

%H Amiram Eldar, <a href="/A290945/b290945.txt">Table of n, a(n) for n = 1..8920</a> (terms below 10^22 calculated using data from Claude Goutier)

%H Claude Goutier, <a href="http://www-labs.iro.umontreal.ca/~goutier/OEIS/A055553/">Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22</a>.

%H Jonathan Vos Post, <a href="https://web.archive.org/web/20200224223330/http://magicdragon.com/fig.html">Table of Figurate Numbers, Sorted, Through 10,000</a>. [Wayback Machine link]

%H <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.

%e 8911 = A000217(133) = A002997(7) therefore 8911 is in the sequence.

%t seqQ[n_]:=IntegerQ[Sqrt[8n+1]] && !PrimeQ[n] && (Mod[n, CarmichaelLambda[n]] == 1); Select[Range[10^6], seqQ]

%Y Cf. A000217, A002817, A002997.

%K nonn

%O 1,1

%A _Amiram Eldar_, Aug 14 2017