A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2*a(n)-1 and 2*a(n)+1 are primes.

2, 3, 6, 9, 15, 21, 30, 36, 51, 54, 69, 75, 90, 96, 99, 114, 120, 135, 141, 156, 174, 210, 216, 231, 261, 285, 300, 309, 321, 330, 405, 411, 414, 429, 441, 510, 516, 525, 531, 546, 576, 615, 639, 645, 651, 660, 714, 726, 741, 744, 804, 810, 834, 849, 861, 894

Offset

1,1

Comments

Intersection of A005097 and A006254. - Zak Seidov, Mar 18 2005

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.

These numbers are never k mod (2k+1) or (k+1) mod (2k+1) with 2k+1 < a(n). - Jon Perry, Sep 04 2012

Excluding the first term, all remaining terms have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013

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

Links

T. D. Noe, Table of n, a(n) for n = 1..10001

Formula

a(n) = A014574(n)/2 = A054735(n+1)/4 = A111046(n+1)/8.

For n > 1, a(n) = 3*A002822(n-1). - Jason Kimberley, Nov 06 2015

A260689(a(n),1) = A264526(a(n)) = 1. - Reinhard Zumkeller, Nov 17 2015

From Michael G. Kaarhus, Aug 19 2022: (Start)

a(n) = (A001359(n) + 1)/2.

a(n) = (A006512(n) - 1)/2.

For n > 1, a(n) = A167379(n-1) * 3/2. (End)

Maple

P := select(isprime, [$1..1789]): map(p->(p+1)/2, select(p->member(p+2, P), P)); # Peter Luschny, Mar 03 2011

Mathematica

Select[Range[900], And @@ PrimeQ[{-1, 1} + 2# ] &] (* Ray Chandler, Oct 12 2005 *)

Prog

(PARI) p=2; forprime(b=3, 1e4, if(b-p==2, print1((p+1)/2", ")); p=b) \\ Altug Alkan, Nov 10 2015
(Haskell)
a040040 = flip div 2 . a014574  -- Reinhard Zumkeller, Nov 17 2015

Crossrefs

Cf. A001359, A006512, A014574, A054735, A111046, A045753 (even terms halved), A002822 (terms divided by 3).

Cf. A221310.

Cf. A260689, A264526.

Sequence in context: A337114 A308870 A273371 * A168497 A256975 A364841

Adjacent sequences: A040037 A040038 A040039 * A040041 A040042 A040043

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Cino Hilliard, Oct 21 2002

Title corrected by Daniel Forgues, Jun 01 2009

Edited by Daniel Forgues, Jun 21 2009

Comment corrected by Daniel Forgues, Jul 12 2009

Status

approved