OFFSET
1,2
COMMENTS
Or, numbers k such that 2k+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. - Jonathan Vos Post, Jun 14 2010
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
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
A193773(a(n)) = 1. - Reinhard Zumkeller, Jan 02 2013
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
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
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
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
Richard R. Forberg, Comments on A005097
Peter Horak and Bader F. AlBdaiwi, Diameter Perfect Lee Codes, arXiv:1109.3475 [cs.IT], 2011-2012.
Daniel Kohen and Ivan Sadofschi, A New Approach on the Seating Couples Problem, arXiv:1006.2571 [math.CO], 2010.
Dhananjay P. Mehendale, On Hamilton Decompositions, arXiv:0806.0251 [math.GM], 2008.
Eric Weisstein's World of Mathematics, Legendre Symbol
FORMULA
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
a(n) = (A065091(n) - 1) / 2. - Reinhard Zumkeller, Jan 02 2013
a(n) ~ n*log(n)/2. - Ilya Gutkovskiy, Jul 11 2016
a(n) = A294507(n) (mod prime(n+1)). - Jonathan Sondow, Nov 04 2017
a(n) = A130290(n+1). - Chai Wah Wu, Jun 04 2022
MAPLE
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
MATHEMATICA
Table[p=Prime[n]; (p-1)/2, {n, 2, 22}] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
(Prime[Range[2, 70]]-1)/2 (* Harvey P. Dale, Jul 11 2020 *)
PROG
(PARI) forprime(p=3, 1e4, print1(p>>1", ")) \\ Charles R Greathouse IV, Jun 16 2011
(Haskell)
a005097 = (`div` 2) . a065091 -- Reinhard Zumkeller, Jan 02 2013
(Magma) [n: n in [1..160] |IsPrime(2*n+1)]; // Vincenzo Librandi, Feb 16 2015
(Python)
from sympy import prime
def A005097(n): return prime(n+1)//2 # Chai Wah Wu, Jun 04 2022
CROSSREFS
A130290 is an essentially identical sequence.
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).
KEYWORD
nonn,easy
AUTHOR
STATUS
approved