This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A040040 #88 Oct 26 2022 19:08:03
%S A040040 2,3,6,9,15,21,30,36,51,54,69,75,90,96,99,114,120,135,141,156,174,210,
%T A040040 216,231,261,285,300,309,321,330,405,411,414,429,441,510,516,525,531,
%U A040040 546,576,615,639,645,651,660,714,726,741,744,804,810,834,849,861,894
%N A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2*a(n)-1 and 2*a(n)+1 are primes.
%C A040040 Intersection of A005097 and A006254. - _Zak Seidov_, Mar 18 2005
%C A040040 The only possible pairs for 2a(n)+-1 are prime/prime (this sequence), not prime/not prime (A104278), prime/notprime (A104279) and not prime/prime (A104280), ... this sequence + A104280 + A104279 + A104278 = the odd numbers.
%C A040040 These numbers are never k mod (2k+1) or (k+1) mod (2k+1) with 2k+1 < a(n). - _Jon Perry_, Sep 04 2012
%C A040040 Excluding the first term, all remaining terms have digital root 3, 6 or 9. - _J. W. Helkenberg_, Jul 24 2013
%C A040040 Positive numbers x such that the difference between x^2 and adjacent squares are prime (both x^2-(x-1)^2 and (x+1)^2-x^2 are prime). - _Doug Bell_, Aug 21 2015
%H A040040 T. D. Noe, <a href="/A040040/b040040.txt">Table of n, a(n) for n = 1..10001</a>
%F A040040 a(n) = A014574(n)/2 = A054735(n+1)/4 = A111046(n+1)/8.
%F A040040 For n > 1, a(n) = 3*A002822(n-1). - _Jason Kimberley_, Nov 06 2015
%F A040040 A260689(a(n),1) = A264526(a(n)) = 1. - _Reinhard Zumkeller_, Nov 17 2015
%F A040040 From _Michael G. Kaarhus_, Aug 19 2022: (Start)
%F A040040 a(n) = (A001359(n) + 1)/2.
%F A040040 a(n) = (A006512(n) - 1)/2.
%F A040040 For n > 1, a(n) = A167379(n-1) * 3/2. (End)
%p A040040 P := select(isprime,[$1..1789]): map(p->(p+1)/2, select(p->member(p+2,P),P)); # _Peter Luschny_, Mar 03 2011
%t A040040 Select[Range[900], And @@ PrimeQ[{-1, 1} + 2# ] &] (* _Ray Chandler_, Oct 12 2005 *)
%o A040040 (PARI) p=2; forprime(b=3, 1e4, if(b-p==2, print1((p+1)/2", ")); p=b) \\ _Altug Alkan_, Nov 10 2015
%o A040040 (Haskell)
%o A040040 a040040 = flip div 2 . a014574 -- _Reinhard Zumkeller_, Nov 17 2015
%Y A040040 Cf. A001359, A006512, A014574, A054735, A111046, A045753 (even terms halved), A002822 (terms divided by 3).
%Y A040040 Cf. A221310.
%Y A040040 Cf. A260689, A264526.
%K A040040 nonn,easy
%O A040040 1,1
%A A040040 _N. J. A. Sloane_
%E A040040 More terms from _Cino Hilliard_, Oct 21 2002
%E A040040 Title corrected by _Daniel Forgues_, Jun 01 2009
%E A040040 Edited by _Daniel Forgues_, Jun 21 2009
%E A040040 Comment corrected by _Daniel Forgues_, Jul 12 2009