|
| |
|
|
A005097
|
|
(Odd primes - 1)/2.
|
|
67
| |
|
|
1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| Or, numbers n such that 2n+1 is prime.
Also numbers not of the form 2xy+x+y. - Jose Brox (tautocrona(AT)terra.es), Dec 29 2005
This sequence arises if you factor the product of a large number of the first odd numbers into the form 3^n(3)5^n(5)7^n(7)11^n(11)... Then n(3)/n(5) = 2, n(3)/n(7) = 3, n(3)/n(11) = 5,... . - Andrzej Staruszkiewicz (astar(AT)th.if.uj.edu.pl), May 31 2007
Kohen shows: A king invites n couples to sit around a round table with 2n+1 seats. For each couple, the king decides a prescribed distance d between 1 and n which the two spouses have to be seated from each other (distance d means that they are separated by exactly d-1 chairs). We will show that there is a solution for every choice of the distances if and only if 2n+1 is a prime number [i.e. iff n is in A005097], using a theorem known as Combinatorial Nullstellensatz. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 14 2010]
Starting from 6, positions at which new primes are seen for Goldbach partitions Eg 31 is first seen at 34 from 31+3, so position=1+(34-6)/2=15 [From Bill R McEachen (bmceache(AT)centralsan.org), Jul 05 2010]
Perfect error-correcting Lee codes of word length n over Z: it is conjectured that these always exist when 2n+1 is a prime, as mentioned in Horak [Jonathan Vos Post, Sep 19 2011].
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
Daniel Kohen and Ivan Sadofschi, A New Approach on the Seating Couples Problem, Jun 13, 2010.
Dhananjay P. Mehendale, On Hamilton Decompositions, arXiv:0806.0251
Peter Horak, Bader F. AlBdaiwi, Diameter Perfect Lee Codes, arXiv:1109.3475v1 [cs.IT], Sep 15, 2011.
|
|
|
FORMULA
| a(n) = A006093(n)/2 = A000010[A000040(n)]/2
|
|
|
MATHEMATICA
| Table[p=Prime[n]; (p-1)/2, {n, 2, 22}] (from Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 29 2008)
|
|
|
PROG
| (PARI) forprime(p=3, 1e4, print1(p>>1", ")) \\ Charles R Greathouse IV, Jun 16 2011
|
|
|
CROSSREFS
| Complement of A047845. Cf. A000040, A006005, A006093.
A130290 is an essentially identical sequence.
Sequence in context: A130290 A161719 * A102781 A139791 A147855 A027563
Adjacent sequences: A005094 A005095 A005096 * A005098 A005099 A005100
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
