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A014574 Average of twin prime pairs. 215
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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). [From Farideh Firoozbakht, Jul 04 2010]

Solutions of the equation (n-1)'+(n+1)'=2, where n' is the arithmetic derivative of n. [Paolo P. Lava, Dec 18 2012]

Subsequence of A217259 - numbers n such that antisigma(n+1) - antisigma(n-1) = 2*n - 1, where antisigma(m) = A024816(m) = sum of nondivisors of m. If n = average of twin prime pairs (q < p) then antisigma(p) - antisigma(q) = 2*n - 1 = p + q - 1. - Jaroslav Krizek, Mar 17 2013

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

REFERENCES

Archimedeans Problems Drive, Eureka, 30 (1967).

Y. Fujiwara, Parsing a Sequence of Qubits, IEEE Trans. Information Theory, 59 (2013), 6796-6806.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

C. K. Caldwell, Twin Primes

C. K. Caldwell, Twin primes

Eric Weisstein's World of Mathematics, Twin Primes

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

a(n) = A141515(k) iff A141515(k) -/+1 are both prime. [From Giovanni Teofilatto, Sep 19 2008]

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

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 *)

PROG

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

(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 */

(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

CROSSREFS

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

Sequence in context: A061715 A072570 A217259 * A034425 A073123 A079865

Adjacent sequences:  A014571 A014572 A014573 * A014575 A014576 A014577

KEYWORD

nonn,easy,nice

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified September 2 02:37 EDT 2014. Contains 246321 sequences.