A286682 a(n) = A059784(n+1) - A059784(n)^2.

1, 4, 12, 4, 22, 12, 114, 4, 138, 142, 2956, 6388, 5248, 17532, 96930, 83782, 1464, 897448, 300832, 26908

Offset: 1

Author

Jeppe Stig Nielsen, May 12 2017

Keywords

• nonn
• hard
• more

Comments

This sequence relates to A059784 just like A108739 relates to the Mills primes A051254.

That this leads to a Mills-like real constant C such that floor(C^2^n) is a prime number for any natural number n, requires a proof of Legendre's conjecture that there is always a prime between consecutive perfect squares.

a(18) and a(19) generate 96042- and 192083-decimal digit probable primes. - Serge Batalov, May 27 2024

a(20) generates a 384166-decimal digit probable prime. - Serge Batalov, May 27 2024

Examples

			A059784(8) by construction can be written ((((((2^2 + 1)^2 + 4)^2 + 12)^2 + 4)^2 + 22)^2 + 12)^2 + 114. Taking out the addends gives 1, 4, 12, 4, 22, 12, 114 which lists the first seven terms of this sequence.
		

Crossrefs

Cf. A059784, A108739, A051254.

Programs

Mathematica

Map[#2 - #1^2 & @@ # &, Partition[NestList[NextPrime[#^2] &, 2, 12], 2, 1]] (* Michael De Vlieger, May 12 2017 *)

PARI

p=2;while(1,a=nextprime(p^2);print1(a-p^2,", ");p=a)

Extensions

a(14)-a(17) from Serge Batalov, May 26 2024

a(18)-a(20) from Serge Batalov, May 27 2024