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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.

%I A293625 #16 Aug 11 2025 08:39:52
%S A293625 1,15,45,56,71,85,121,141,155,176,185,206,255,275,301,346,350,380,401,
%T A293625 470,506,511,540,680,710,745,786,801,871,946,1025,1156,1200,1211,1326,
%U A293625 1380,1395,1421,1480,1505,1515,1590,1676,1696,1710,1830,1941,2066,2171
%N 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.
%C A293625 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).
%C A293625 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.
%H A293625 Amiram Eldar, <a href="/A293625/b293625.txt">Table of n, a(n) for n = 1..10000</a>
%H A293625 Andrzej Rotkiewicz, <a href="https://gdz.sub.uni-goettingen.de/id/PPN378850199_0028">On pyramidal numbers of order 4</a>, Elemente der Mathematik, Vol. 28 (1973), pp. 14-16.
%e A293625 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.
%t A293625 Select[Range[1, 1000], PrimeQ[12#+1] && PrimeQ[18#+1] && PrimeQ[36#+1] &]
%Y A293625 Cf. A000330, A001567, A293624.
%K A293625 nonn
%O A293625 1,2
%A A293625 _Amiram Eldar_, Oct 13 2017