A152786 Integers k such that (k^2)/2 is the arithmetic mean of a pair of twin primes.

6, 12, 42, 48, 72, 84, 90, 174, 204, 264, 306, 372, 408, 456, 474, 546, 594, 600, 642, 750, 852, 882, 936, 972, 978, 1038, 1140, 1212, 1272, 1386, 1470, 1512, 1518, 1584, 1770, 1836, 1902, 1980, 1986, 2130, 2196, 2256, 2262, 2316, 2382, 2652, 2688, 2718

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Dec 12 2008

Keywords

• nonn

Comments

Square roots of A054735 where these are integer.

Examples

			6 is a term since (6^2)/2 = 18 = mean(17, 19).
12 is a term since (12^2)/2 = 72 = mean(71,73).
42 is a term since (42^2)/2 = 882 = mean(881,883).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..4288 from Zak Seidov)
• Zak Seidov, A152786 = 6*A037073: near-duplicates?, seqfan list, Aug 20 2010.

Crossrefs

Cf. A014574, A037073, A054735, A152788 (cubic version).

Subsequence of A074924. - Zak Seidov, Feb 01 2013

Programs

Magma

[k:k in [2..2800 by 2]| IsPrime(k*k div 2 -1) and IsPrime(k*k div 2 +1)]; // Marius A. Burtea, Jan 01 2020

Maple

isa := n -> isprime(n) and isprime(n+2) and issqr(2*n+2):
select(isa, [$4..1000000]): map(n -> sqrt(2*n+2), %); # Peter Luschny, Jan 05 2020

Mathematica

lst={};Do[p1=Prime[n];p2=Prime[n+1];If[p2-p1==2,e=(2*(p1+1))^(1/2);i=Floor[e]; If[e==i,AppendTo[lst,i]]],{n,3*9!}];lst
(* Second program: *)
Select[Map[Sqrt[2 #] &, Mean /@ Select[Partition[Prime@ Range[10^6], 2, 1], Subtract @@ # == -2 &]], IntegerQ] (* Michael De Vlieger, Feb 18 2018 *)

PARI

forstep(n=6,1e3,6,if(isprime(n^2/2-1)&&isprime(n^2/2+1),print1(n", "))) \\ Charles R Greathouse IV, Feb 01 2013

Formula

{n: n^2 = A054735(i), any i}. - R. J. Mathar, Dec 12 2008

a(n) = 6*A037073(n). [Zak Seidov, seqfan list, Aug 20 2010] [From R. J. Mathar, Sep 07 2010]

Extensions

Edited by R. J. Mathar, Dec 12 2008