A365050 Slowest increasing sequence of primes such that a(n - 1) + a(n) and a(n - 1)^2 + a(n)^2 are both semiprimes, with a(1)=2.

2, 19, 1459, 1699, 3079, 3259, 5419, 5479, 6079, 6679, 7219, 8059, 8719, 11299, 12619, 13219, 13399, 15559, 15679, 18919, 24379, 25219, 26839, 34819, 38239, 39019, 39799, 40459, 40759, 42019, 43399, 44119, 47059, 47779, 54559, 55339, 57139, 60259, 65479, 65599, 68659, 69859, 72559, 77659, 78439

Offset

1,1

Comments

For n > 1, (a(n - 1) + a(n))/2 and (a(n - 1)^2 + a(n)^2)/2 are primes and a(n) == 19 (mod 60).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(2) = 19 because 2+19=21=3*7 and 2^2+19^2=365=3*73 are semiprimes, and none of the primes from 3 to 17 works.

Maple

R:= 2, 19: p:= 19: count:= 1: q:= 19:
while count < 100 do
  q:= nextprime(q);
  if isprime((p+q)/2) and isprime((p^2+q^2)/2) then
    R:= R, q; p:= q; count:= count+1;
  fi
od:
R;

Mathematica

s = {2}; p = 2; Do[q = NextPrime[p]; While[{2, 2} != PrimeOmega[{p + q, p^2 + q^2}],  q = NextPrime[q]]; AppendTo[s, p = q], {10}]; s

Crossrefs

Cf. A001358.

Sequence in context: A369946 A350933 A172028 * A024229 A094663 A274762

Adjacent sequences: A365047 A365048 A365049 * A365051 A365052 A365053

Keyword

nonn

Author

Zak Seidov and Robert Israel, Aug 18 2023

Status

approved