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). - 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
FORMULA
a(n) = A129297(n+4). - Reinhard Zumkeller, Apr 09 2007
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
(GAP) a:=1+Filtered([1..2000], p->IsPrime(p) and IsPrime(p+2)); # Muniru A Asiru, May 20 2018
CROSSREFS
KEYWORD
nonn,easy,nice,changed
AUTHOR
EXTENSIONS
Offset changed to 1 by R. J. Mathar, Jun 11 2011
STATUS
approved