A266409 -id:A266409 - cp's OEIS frontend

A266418 Permutation of natural numbers: a(1) = 1, a(2n) = A266410(a(n)), a(2n+1) = A266409(1+a(n)).

1, 4, 2, 10, 6, 7, 3, 22, 20, 15, 11, 16, 12, 9, 5, 40, 53, 37, 45, 29, 33, 24, 21, 31, 35, 25, 23, 19, 18, 13, 8, 68, 111, 85, 156, 64, 104, 75, 123, 51, 74, 56, 87, 43, 59, 39, 48, 54, 80, 61, 90, 46, 60, 42, 57, 36, 44, 34, 41, 27, 26, 17, 14, 107, 210, 167, 387, 133, 276, 229, 573, 101, 198, 158, 351, 120

Offset: 1

Antti Karttunen, Jan 28 2016

This sequence can be represented as a binary tree. Each left hand child is produced as A266410(n), and each right hand child as A266409(1+n), when the parent node contains n:
                                     |
                  ...................1...................
                 4                                       2
      10......../ \........6                   7......../ \........3
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  22       20         15       11         16       12          9       5
40  53   37  45     29  33   24  21     31  35   25  23      19 18   13 8
etc.

Antti Karttunen, Table of n, a(n) for n = 1..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A266417.
Cf. A266409, A266410.
Similar or related permutations: A237126, A266638.

a(1) = 1, after which: a(2n) = A266410(a(n)), a(2n+1) = A266409(1+a(n)).
As a composition of related permutations:
a(n) = A266638(A237126(n)).

A266350 Partial sums of f(n) = A192490(2n+1); least monotonic left inverse of A266409.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 9, 9, 9, 10, 10, 11, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 21, 21, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 25, 26, 26, 26, 27, 27, 28, 28, 28, 28, 28, 28, 29, 29, 29, 30, 30, 30, 30, 31, 31, 32, 32

Offset: 1

Antti Karttunen, Jan 28 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10011

Cf. A192490, A266409.

Cf. also permutations A266417, A266637.

a(1) = 1; for n > 1, a(n) = a(n-1) + A192490(2n + 1).
Other identities. For all n >= 1:
a(A266409(n)) = n.

A005097 (Odd primes - 1)/2.

Original entry on oeis.org

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

N. J. A. Sloane

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

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

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.

a005097 = (`div` 2) . a065091 -- Reinhard Zumkeller, Jan 02 2013

[n: n in [1..160] |IsPrime(2*n+1)]; // Vincenzo Librandi, Feb 16 2015

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

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 *)

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

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

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

A266410 a(n) = (A266419(n) - 1) / 2; numbers n such that 2n+1 is a nonludic number (in A192607).

Original entry on oeis.org

4, 7, 9, 10, 13, 15, 16, 17, 19, 22, 24, 25, 27, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 43, 46, 47, 49, 50, 51, 52, 54, 55, 56, 58, 61, 62, 64, 66, 67, 68, 69, 70, 72, 73, 75, 76, 77, 79, 81, 82, 83, 84, 85, 88, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 106, 107, 108, 109, 112, 114, 115, 118, 120, 121

Offset: 1

Antti Karttunen, Jan 28 2016

nonn

Odd nonludic numbers (A266419) decremented by one, then halved.

Antti Karttunen, Table of n, a(n) for n = 1..10220

Complement: A266409.
Cf. A192490, A192607, A266419.
Cf. permutations A266418, A266638.
Cf. also A047845.

a(n) = (A266419(n) - 1) / 2.

A266638 a(1) = 1, a(ludic(n)) = (ludic(3+a(n-1))-1)/2, a(nonludic(n)) = A266410(a(n)), where ludic(n) = n-th ludic number A003309, nonludic(n) = n-th nonludic number A192607 and A266410 = numbers n such that 2n+1 is nonludic.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 9, 10, 13, 8, 16, 12, 15, 19, 22, 11, 27, 17, 31, 25, 29, 18, 36, 20, 40, 24, 49, 26, 32, 54, 46, 51, 34, 62, 37, 14, 68, 43, 81, 35, 47, 23, 55, 88, 76, 33, 83, 58, 99, 64, 28, 44, 107, 72, 127, 61, 77, 42, 91, 53, 136, 121, 56, 130, 94, 21, 151, 101, 50, 65, 73, 161, 114, 189, 98, 38

Offset: 1

Antti Karttunen, Jan 28 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A266637.
Cf. A003309, A192490, A192512, A236863, A266409, A266410.
Related or similar permutations: A237427, A266418.

a(1) = 1; for n > 1, if A192490(n) = 1 [when n is one of Ludic numbers, A003309] a(n) = A266409(1+a(A192512(n)-1)), otherwise a(n) = A266410(a(A236863(n))).
As a composition of related permutations:
a(n) = A266418(A237427(n)).

cp's OEIS Frontend

A266418 Permutation of natural numbers: a(1) = 1, a(2n) = A266410(a(n)), a(2n+1) = A266409(1+a(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A266350 Partial sums of f(n) = A192490(2n+1); least monotonic left inverse of A266409.

Views

Author

Keywords

Links

Crossrefs

Formula

A005097 (Odd primes - 1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

A266410 a(n) = (A266419(n) - 1) / 2; numbers n such that 2n+1 is a nonludic number (in A192607).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A266638 a(1) = 1, a(ludic(n)) = (ludic(3+a(n-1))-1)/2, a(nonludic(n)) = A266410(a(n)), where ludic(n) = n-th ludic number A003309, nonludic(n) = n-th nonludic number A192607 and A266410 = numbers n such that 2n+1 is nonludic.

Views

Author

Keywords

Links

Crossrefs

Formula