A235592 Numbers k such that k*(k+1) - prime(k) is prime.

2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 18, 19, 20, 21, 26, 27, 29, 34, 36, 37, 38, 41, 44, 45, 48, 53, 54, 57, 61, 62, 69, 70, 71, 85, 86, 87, 89, 90, 98, 99, 102, 105, 112, 114, 117, 119, 131, 134, 135, 136, 137, 141, 145, 147, 149, 150, 153, 156, 157, 162, 170, 171, 175, 176, 180, 183, 187, 189, 198, 200

Offset: 1

Author

Zhi-Wei Sun, Jan 12 2014

Keywords

• nonn

Comments

It is known that prime(k) <= k*(k+1) for any positive integer k. The conjecture in A235613 implies that the sequence has infinitely many terms.

Conjecture: This sequence contains infinitely many primes.

Examples

			a(1) = 2 since 1*2 - prime(1) = 0 is not prime, but 2*3 - prime(2) = 3 is prime.
a(2) = 3 since 3*4 - prime(3) = 7 is prime.
a(3) = 4 since 4*5 - prime(4) = 13 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A235613, A235614.

Programs

Mathematica

n=0;Do[If[PrimeQ[k(k+1)-Prime[k]],n=n+1;Print[n," ",k]],{k,1,200}]
Select[Range[200],PrimeQ[(#(#+1))-Prime[#]]&] (* Harvey P. Dale, Apr 10 2020 *)