A014574 Average of twin prime pairs.

4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488, 1608

Offset: 1

Author

R. K. Guy, N. J. A. Sloane, Eric W. Weisstein

Keywords

• nonn
• easy
• nice

Comments

With an initial 1 added, this is the complement of the closure of {2} under a*b+1 and a*b-1. - Franklin T. Adams-Watters, Jan 11 2006

Also the square root of the product of twin prime pairs + 1. Two consecutive odd numbers can be written as 2k+1,2k+3. Then (2k+1)(2k+3)+1 = 4(k^2+2k+1) = 4(k+1)^2, a perfect square. Since twin prime pairs are two consecutive odd numbers, the statement is true for all twin prime pairs. - Cino Hilliard, May 03 2006

Or, single (or isolated) composites. Nonprimes k such that neither k-1 nor k+1 is nonprime. - Juri-Stepan Gerasimov, Aug 11 2009

Numbers n such that sigma(n-1) = phi(n+1). - Farideh Firoozbakht, Jul 04 2010

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

Numbers n such that n^2-1 is a semiprime. - Thomas Ordowski, Sep 24 2015

Every term but the first is a multiple of 6. - Harvey P. Dale, Mar 31 2023

References

• Archimedeans Problems Drive, Eureka, 30 (1967).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• C. K. Caldwell, The Prime Glossary: Twin primes
• C. K. Caldwell, The Top Twenty: Twin Primes
• Y. Fujiwara, Parsing a Sequence of Qubits, IEEE Trans. Information Theory, 59 (2013), 6796-6806.
• Y. Fujiwara, Parsing a Sequence of Qubits, arXiv:1207.1138 [quant-ph], 2012-2013.
• L. J. Gerstein, A reformulation of the Goldbach conjecture, Math. Mag., 66 (1993), 44-45.
• Brian Hayes, Does having prime neighbors make you more composite?, Bit-Player Article, Nov 04 2021
• Eric Weisstein's World of Mathematics, Twin Primes

Crossrefs

Cf. A000010, A000203, A001359, A002822, A006512, A037074, A040040, A054735, A077800, A111046.

A068507 is the intersection of A002182 and this sequence.

Programs

GAP

a:=1+Filtered([1..2000],p->IsPrime(p) and IsPrime(p+2)); # Muniru A Asiru, May 20 2018

Haskell

a014574 n = a014574_list !! (n-1)
a014574_list = [x | x <- [2,4..], a010051 (x-1) == 1, a010051 (x+1) == 1]
-- Reinhard Zumkeller, Apr 11 2012

Maple

P := select(isprime,[$1..1609]): map(p->p+1,select(p->member(p+2,P),P)); # Peter Luschny, Mar 03 2011
A014574 := proc(n) option remember; local p ; if n = 1 then 4 ; else p := nextprime( procname(n-1) ) ; while not isprime(p+2) do p := nextprime(p) ; od ; return p+1 ; end if ; end proc: # R. J. Mathar, Jun 11 2011

Mathematica

Select[Table[Prime[n] + 1, {n, 260}], PrimeQ[ # + 1] &] (* Ray Chandler, Oct 12 2005 *)
Mean/@Select[Partition[Prime[Range[300]],2,1],Last[#]-First[#]==2&] (* Harvey P. Dale, Jan 16 2014 *)

Maxima

A014574(n) := block(
    if n = 1 then
        return(4),
    p : A014574(n-1) ,
    for k : 2 step 2 do (
        if primep(p+k-1) and primep(p+k+1) then
            return(p+k)
    )
)$ /* R. J. Mathar, Mar 15 2012 */

PARI

p=2;forprime(q=3,1e4,if(q-p==2,print1(p+1", "));p=q) \\ Charles R Greathouse IV, Jun 10 2011

Formula

a(n) = (A001359(n) + A006512(n))/2 = 2*A040040(n) = A054735(n)/2 = A111046(n)/4.

a(n) = A129297(n+4). - Reinhard Zumkeller, Apr 09 2007

A010051(a(n) - 1) * A010051(a(n) + 1) = 1. Reinhard Zumkeller, Apr 11 2012

a(n) = 6*A002822(n-1), n>=2. - Ivan N. Ianakiev, Aug 19 2013

a(n)^4 - 4*a(n)^2 = A062354(a(n)^2 - 1). - Raphie Frank, Oct 17 2013

Extensions

Offset changed to 1 by R. J. Mathar, Jun 11 2011