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A152786 Integers k such that (k^2)/2 is the arithmetic mean of a pair of twin primes. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Square roots of A054735 where these are integer.

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.

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]

EXAMPLE

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).

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 *)

PROG

(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

(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

CROSSREFS

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

Subsequence of A074924. - Zak Seidov, Feb 01 2013

Sequence in context: A307181 A052747 A007121 * A267309 A206039 A048069

Adjacent sequences:  A152783 A152784 A152785 * A152787 A152788 A152789

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 12 2008

EXTENSIONS

Edited by R. J. Mathar, Dec 12 2008

STATUS

approved

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Last modified March 9 02:20 EST 2021. Contains 341961 sequences. (Running on oeis4.)