A293623 Fermat pseudoprimes to base 2 that are pentagonal.

7957, 241001, 1419607, 1830985, 1993537, 2134277, 2163001, 2491637, 2977217, 4864501, 5351537, 6952037, 10084177, 11367137, 11433301, 14609401, 21306157, 22591301, 26470501, 26977001, 29581501, 35851037, 44731051, 46517857, 53154337, 55318957, 55610837

Offset: 1

Author

Amiram Eldar, Oct 13 2017

Keywords

• nonn

Comments

Rotkiewicz proved that this sequence is infinite.

Intersection of A001567 and A000326.

The corresponding indices of the pentagonal numbers are 73, 401, 973, 1105, 1153, 1193, 1201, 1289, 1409, 1801, 1889, 2153, 2593, 2753, 2761, ...

Examples

			7957 = (3*73^2 - 73)/2 is in the sequence since it is pentagonal, composite, and 2^7956 == 1 (mod 7957).
		

References

• Andrzej Rotkiewicz, Sur les nombres pseudopremiers pentagonaux, Bull. Soc. Roy. Sci. Liège, Vol. 33 (1964), pp. 261-263.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000326, A001567.

Programs

Mathematica

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