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A014574
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Average of twin prime pairs.
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368
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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
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OFFSET
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1,1
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COMMENTS
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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
Solutions of the equation (n-1)'+(n+1)'=2, where n' is the arithmetic derivative of n. - Paolo P. Lava, Dec 18 2012
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
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REFERENCES
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Archimedeans Problems Drive, Eureka, 30 (1967).
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LINKS
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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
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FORMULA
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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
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MAPLE
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P := select(isprime, [$1..1609]): map(p->p+1, select(p->member(p+2, P), P)); # Peter Luschny, Mar 03 2011
A014574 := proc(n) option remember; local p ; if n = 1 then 4 ; else p := nextprime( procname(n-1) ) ; while not isprime(p+2) do p := nextprime(p) ; od ; return p+1 ; end if ; end proc: # R. J. Mathar, Jun 11 2011
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MATHEMATICA
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Select[Table[Prime[n] + 1, {n, 260}], PrimeQ[ # + 1] &] (* Ray Chandler, Oct 12 2005 *)
Mean/@Select[Partition[Prime[Range[300]], 2, 1], Last[#]-First[#]==2&] (* Harvey P. Dale, Jan 16 2014 *)
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PROG
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(PARI) p=2; forprime(q=3, 1e4, if(q-p==2, print1(p+1", ")); p=q) \\ Charles R Greathouse IV, Jun 10 2011
(Maxima) A014574(n) := block(
if n = 1 then
return(4),
p : A014574(n-1) ,
for k : 2 step 2 do (
if primep(p+k-1) and primep(p+k+1) then
return(p+k)
)
)$ /* 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
(GAP) a:=1+Filtered([1..2000], p->IsPrime(p) and IsPrime(p+2)); # Muniru A Asiru, May 20 2018
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CROSSREFS
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Cf. A000010, A000203, A001359, A002822, A006512, A037074, A040040, A054735, A077800, A111046.
A068507 is the intersection of A002182 and this sequence.
Sequence in context: A353073 A072570 A217259 * A258838 A034425 A073123
Adjacent sequences: A014571 A014572 A014573 * A014575 A014576 A014577
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KEYWORD
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nonn,easy,nice
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AUTHOR
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R. K. Guy, N. J. A. Sloane, Eric W. Weisstein
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EXTENSIONS
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Offset changed to 1 by R. J. Mathar, Jun 11 2011
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STATUS
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approved
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