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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Crossrefs

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

Cf. A221310.

Cf. A260689, A264526.

Programs

Haskell

a040040 = flip div 2 . a014574  -- Reinhard Zumkeller, Nov 17 2015

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 *)

PARI

p=2; forprime(b=3, 1e4, if(b-p==2, print1((p+1)/2", ")); p=b) \\ Altug Alkan, Nov 10 2015

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)

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