A293625 Generators of Fermat pseudoprimes to base 2 that are square pyramidal numbers: numbers k such that 12*k+1, 18*k+1 and 36*k+1 are all primes.

1, 15, 45, 56, 71, 85, 121, 141, 155, 176, 185, 206, 255, 275, 301, 346, 350, 380, 401, 470, 506, 511, 540, 680, 710, 745, 786, 801, 871, 946, 1025, 1156, 1200, 1211, 1326, 1380, 1395, 1421, 1480, 1505, 1515, 1590, 1676, 1696, 1710, 1830, 1941, 2066, 2171

Offset: 1

Author

Amiram Eldar, Oct 13 2017

Keywords

• nonn

Comments

Rotkiewicz proved that if n is in the sequence then P((2^(2(18n+1))-1)/3) is a square pyramidal Fermat pseudoprime to base 2, where P(k) = k*(k+1)*(2k+1)/6 (A000330).

The generated numbers are terms in A293624. The first term is 256409721410526509996425240557391, the next 2 terms are about 3.683...*10^487 and 8.007...*10^1462.

Examples

			1 is in the sequence since 12*1+1 = 13, 18*1+1 = 19 and 36*1+1 = 37 are all primes. P((2^(2(18*1+1))-1)/3) = P(91625968981) = 256409721410526509996425240557391 is a Fermat pseudoprime to base 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Andrzej Rotkiewicz, On pyramidal numbers of order 4, Elemente der Mathematik, Vol. 28 (1973), pp. 14-16.

Crossrefs

Cf. A000330, A001567, A293624.

Programs

Mathematica

Select[Range[1, 1000], PrimeQ[12#+1] && PrimeQ[18#+1] && PrimeQ[36#+1] &]