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

1,2

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.

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.

Example

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.

Mathematica

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

Crossrefs

Cf. A000330, A001567, A293624.

Sequence in context: A029827 A357706 A119123 * A084821 A345654 A298462

Adjacent sequences: A293622 A293623 A293624 * A293626 A293627 A293628

Keyword

nonn

Author

Amiram Eldar, Oct 13 2017

Status

approved