A365235 Least increasing sequence of primes such that a(n-1)^2 + a(n)^2 is semiprime, with a(1)=2.

2, 19, 29, 59, 71, 79, 101, 131, 149, 151, 191, 251, 281, 331, 379, 389, 401, 449, 461, 499, 509, 521, 569, 571, 599, 641, 659, 691, 739, 761, 811, 919, 971, 991, 1009, 1019, 1129, 1151, 1259, 1321, 1409, 1511, 1531, 1559, 1579, 1601, 1621, 1669, 1699, 1811, 1901, 1931, 1979, 1999, 2081, 2141

Offset

1,1

Comments

For n >= 2, a(n) == 1 or 9 (mod 10) and a(n)^2 + a(n+1)^2 is twice a prime.

Links

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

Example

a(2) = 19 because a(1) = 2 and 2^2 + 19^2 = 365 = 5 * 73 is a semiprime.
a(3) = 29 because 19^2 + 29^2 = 1202 = 2*601 is a semiprime.

Maple

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

Mathematica

p = 3; s = {q = 2}; Do[While[2 != PrimeOmega[q^2 + p^2], p = NextPrime[p]]; AppendTo[s, q = p], {100}]; s

Crossrefs

Cf. A001358.

Sequence in context: A031030 A083689 A102617 * A290163 A284496 A120276

Adjacent sequences: A365232 A365233 A365234 * A365236 A365237 A365238

Keyword

nonn

Author

Zak Seidov and Robert Israel, Aug 28 2023

Status

approved