A290945 Triangular Carmichael numbers.

561, 8911, 10585, 41041, 115921, 314821, 334153, 6313681, 8134561, 14913991, 32914441, 60957361, 67902031, 135556345, 289766701, 321197185, 329769721, 368113411, 471905281, 765245881, 842202361, 962442001, 1507746241, 2489462641, 2588653081, 3104207821

Offset: 1

Author

Amiram Eldar, Aug 14 2017

Keywords

• nonn

Comments

Intersection of A000217 and A002997.

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

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, ...

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).

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..8920 (terms below 10^22 calculated using data from Claude Goutier)
• Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
• Jonathan Vos Post, Table of Figurate Numbers, Sorted, Through 10,000. [Wayback Machine link]
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Cf. A000217, A002817, A002997.

Programs

Mathematica

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