A005097 (Odd primes - 1)/2.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Also solutions to: A000010(2*n+1) = n * A000005(2*n+1). - Enrique Pérez Herrero, Jun 07 2012

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

Crossrefs

Complement of A047845. Cf. A000040, A006005, A006093.

A130290 is an essentially identical sequence.

Cf. A005384 (subsequence of primes), A266400 (their indices in this 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).

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).

Cf. also A266409, A294507.

Programs

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

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 *)

PARI

forprime(p=3,1e4,print1(p>>1", ")) \\ Charles R Greathouse IV, Jun 16 2011

Python

from sympy import prime
def A005097(n): return prime(n+1)//2 # Chai Wah Wu, Jun 04 2022

Formula

a(n) = A006093(n)/2 = A000010(A000040(n+1))/2.

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