A265006 - OEIS

%I #61 May 07 2024 08:01:27

%S 5,7,11,13,29,31,41,43,71,73,239,241,419,421,461,463,599,601,1481,

%T 1483,1721,1723,2549,2551,2969,2971,3539,3541,4421,4423,8009,8011,

%U 10301,10303,17291,17293,19181,19183,20021,20023,23561,23563,24179,24181,27059,27061,31151,31153,35531,35533

%N Twin prime pairs of the form (k^2 + k - 1, k^2 + k + 1).

%C This is a subset of A002327 and A002383 taken together. Note that 3 is not a member, as the pairing (3, 5) is excluded as defined, as 3 and 5 associate to different centers.

%C The corresponding n are in A088485.

%C The average of each twin prime pair is an oblong number (A002378). - _Michel Marcus_, Feb 04 2017

%H Amiram Eldar, <a href="/A265006/b265006.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..3594 from G. C. Greubel)

%F a(2n-1) = A088486(n). a(2n)=2+a(2n-1).

%e For k = 6, k^2 + k = 6^2 + 6 = 42, and (41,43) is a twin prime pair, so 41 and 43 are in the sequence.

%t {#^2 + # - 1, #^2 + # + 1} & /@ Select[Range@ 200, PrimeQ[#^2 + # - 1] && PrimeQ[#^2 + # + 1] &] // Flatten (* _Michael De Vlieger_, Nov 30 2015 *)

%t Flatten[Select[Table[n^2 + n + {-1, 1}, {n, 0, 200}], And@@PrimeQ[#] &]] (* _Vincenzo Librandi_, Feb 05 2017 *)

%o (PARI) genit()={my(maxx=1000);n=0;while(n<maxx,n+=1;q=n^2+n;if( isprime(q-1)&&isprime(q+1),print1(q-1,",",q+1,",")));}

%o (Magma) &cat[[n^2+n-1, n^2+n+1]: n in [0..250]| IsPrime(n^2+n-1) and IsPrime(n^2+n+1)]; // _Vincenzo Librandi_, Feb 05 2017

%Y Cf. A002327, A002378, A002383, A088485, A088486, A306889.

%K nonn,easy,tabf

%O 1,1

%A _Bill McEachen_, Nov 29 2015