A293622 Fermat pseudoprimes to base 2 that are triangular.

561, 2701, 4371, 8911, 10585, 18721, 33153, 41041, 49141, 93961, 104653, 115921, 157641, 226801, 289941, 314821, 334153, 534061, 665281, 721801, 831405, 873181, 915981, 1004653, 1373653, 1537381, 1755001, 1815465, 1987021, 2035153, 2233441, 2284453, 3059101

Offset: 1

Author

Amiram Eldar, Oct 13 2017

Keywords

• nonn

Comments

Rotkiewicz proved that this sequence is infinite.

Intersection of A001567 and A000217.

Supersequence of A290945 (triangular Carmichael numbers).

All values of A098025(n)*(2*A098025(n)-1) are terms in this sequence.

The corresponding indices of the triangular numbers are 33, 73, 93, 133, 145, 193, 257, 286, 313, 433, 457, 481, 561, 673, 761, 793, 817, ...

Examples

			2701 = 73 * 74 / 2 = 37 * 73 is in the sequence since it is triangular and composite, and 2^2700 == 1 (mod 2701).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.
• Andrzej Rotkiewicz, Sur les nombres pseudopremiers triangulaires, Elemente der Mathematik, Vol. 19 (1964), pp. 82-83.

Crossrefs

Cf. A000217, A001567, A098025, A290945.

Programs

Mathematica

t[n_]:=n(n+1)/2; Select[t[Range[3, 10^4]], PowerMod[2, (# - 1), # ] == 1 &]

PARI

forcomposite(c=1, 31*10^5, if(Mod(2, c)^(c-1)==1 && ispolygonal(c, 3), print1(c, ", "))) \\ Felix Fröhlich, Oct 14 2017