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%I #177 Feb 05 2024 00:53:17
%S 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,
%T 51,53,54,56,63,65,68,69,74,75,78,81,83,86,89,90,95,96,98,99,105,111,
%U 113,114,116,119,120,125,128,131,134,135,138,140,141,146,153,155,156
%N (Odd primes - 1)/2.
%C Or, numbers k such that 2k+1 is prime.
%C Also numbers not of the form 2xy + x + y. - Jose Brox (tautocrona(AT)terra.es), Dec 29 2005
%C 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
%C 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. - _Jonathan Vos Post_, Jun 14 2010
%C Starting from 6, positions at which new primes are seen for Goldbach partitions. E.g., 31 is first seen at 34 from 31+3, so position = 1 + (34-6)/2 = 15. - _Bill McEachen_, Jul 05 2010
%C 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
%C Also solutions to: A000010(2*n+1) = n * A000005(2*n+1). - _Enrique Pérez Herrero_, Jun 07 2012
%C A193773(a(n)) = 1. - _Reinhard Zumkeller_, Jan 02 2013
%C I conjecture that the set of pairwise sums of terms of this sequence (A005097) is the set of integers greater than 1, i.e.: 1+1=2, 1+2=3, ..., 5+5=10, ... (This is equivalent to Goldbach's conjecture: every even integer greater than or equal to 6 can be expressed as the sum of two odd primes.) - _Lear Young_, May 20 2014
%C See conjecture and comments from Richard R. Forberg, in Links section below, on the relationship of this sequence to rules on values of c that allow both p^q+c and p^q-c to be prime, for an infinite number of primes p. - _Richard R. Forberg_, Jul 13 2016
%C The sequence represents the minimum number Ng of gears which are needed to draw a complete graph of order p using a Spirograph(R), where p is an odd prime. The resulting graph consists of Ng hypotrochoids whose respective nodes coincide. If the teethed ring has a circumference p then Ng = (p-1)/2. Examples: A complete graph of order three can be drawn with a Spirograph(R) using a ring with 3n teeth and one gear with n teeth. n is an arbitrary number, only related to the geometry of the gears. A complete graph of order 5 can be drawn using a ring with diameter 5 and 2 gears with diameters 1 and 2 respectively. A complete graph of order 7 can be drawn using a ring with diameter 7 and 3 gears with diameters 1, 2 and 3 respectively. - _Bob Andriesse_, Mar 31 2017
%H T. D. Noe, <a href="/A005097/b005097.txt">Table of n, a(n) for n = 1..1000</a>
%H Richard R. Forberg, <a href="/A005097/a005097.txt">Comments on A005097</a>
%H Peter Horak and Bader F. AlBdaiwi, <a href="http://arxiv.org/abs/1109.3475">Diameter Perfect Lee Codes</a>, arXiv:1109.3475 [cs.IT], 2011-2012.
%H Daniel Kohen and Ivan Sadofschi, <a href="http://arxiv.org/abs/1006.2571">A New Approach on the Seating Couples Problem</a>, arXiv:1006.2571 [math.CO], 2010.
%H Dhananjay P. Mehendale, <a href="http://arxiv.org/abs/0806.0251">On Hamilton Decompositions</a>, arXiv:0806.0251 [math.GM], 2008.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendreSymbol.html">Legendre Symbol</a>
%F a(n) = A006093(n)/2 = A000010(A000040(n+1))/2.
%F a(n) = (prime(n+1)^2-1)/(2*sigma(prime(n+1))) = (A000040(n+1)^2-1)/(2*A000203(A000040(n+1))). - _Gary Detlefs_, May 02 2012
%F a(n) = (A065091(n) - 1) / 2. - _Reinhard Zumkeller_, Jan 02 2013
%F a(n) ~ n*log(n)/2. - _Ilya Gutkovskiy_, Jul 11 2016
%F a(n) = A294507(n) (mod prime(n+1)). - _Jonathan Sondow_, Nov 04 2017
%F a(n) = A130290(n+1). - _Chai Wah Wu_, Jun 04 2022
%p with(numtheory): p:=n-> ithprime(n):seq((p(n+1)^2-1)/(2*sigma(p(n+1))), n= 1..64) # _Gary Detlefs_, May 02 2012
%t Table[p=Prime[n];(p-1)/2, {n, 2, 22}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 29 2008 *)
%t (Prime[Range[2,70]]-1)/2 (* _Harvey P. Dale_, Jul 11 2020 *)
%o (PARI) forprime(p=3,1e4,print1(p>>1", ")) \\ _Charles R Greathouse IV_, Jun 16 2011
%o (Haskell)
%o a005097 = (`div` 2) . a065091 -- _Reinhard Zumkeller_, Jan 02 2013
%o (Magma) [n: n in [1..160] |IsPrime(2*n+1)]; // _Vincenzo Librandi_, Feb 16 2015
%o (Python)
%o from sympy import prime
%o def A005097(n): return prime(n+1)//2 # _Chai Wah Wu_, Jun 04 2022
%Y Complement of A047845. Cf. A000040, A006005, A006093.
%Y A130290 is an essentially identical sequence.
%Y Cf. A005384 (subsequence of primes), A266400 (their indices in this sequence).
%Y Numbers n such that 2n+k is prime: this seq(k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
%Y Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).
%Y Cf. also A266409, A294507.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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