A358482 a(n) is the first prime p such that, if q is the next prime, (p*q+p+q)/5^n is a prime.

2, 7, 1847, 90793, 139313, 1790293, 3834043, 5521543, 24996487, 2062865293, 5555052793, 12111965183, 95460776921, 6045070151921, 10204150316653, 70501997496487, 442748358250633, 368313674465183, 2935956099058987, 10360552690003447, 120999670013476223, 1820610211470152737

Offset: 0

Author

Robert Israel, Dec 25 2022

Keywords

• nonn

Comments

Suggested in an email by J. M. Bergot.

For n >= 1, a(n) has the form k * 5^n + x, for some k >= 0, where x is a solution to the modular quadratic equation x^2 + (d+2)*x + d == 0 (mod 5^n), where d = q-p. - Daniel Suteu, Dec 28 2022

Examples

			a(2) = 1847 because 1847 is prime, the next prime is 1861, 1847*1861 + 1847 + 1861 = 3440975 = 5^2 * 137639 where 137639 is prime, and no smaller prime works.
		

Crossrefs

Cf. A000351, A002386, A126148.

Programs

Maple

V:= Array(0..8):
q:= 2: count:= 0:
while count < 9 do
  p:= q; q:= nextprime(q);
t:= p*q+p+q;
k:= padic:-ordp(t,5);
if V[k] = 0 and isprime(t/5^k) then
  V[k]:= p; count:= count+1
fi
od:
convert(V,list);

Extensions

a(11) from Michael S. Branicky, Dec 26 2022

a(12)-a(15) from David A. Corneth, Dec 26 2022

a(16) from Martin Ehrenstein, Dec 27 2022

a(17)-a(21) from Daniel Suteu, Dec 28 2022