A065305 -id:A065305 - cp's OEIS frontend

A065306 The Goldbach permutation: take A065305, cross out repetitions and subtract 2 from each term.

1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13, 15, 14, 16, 17, 18, 19, 21, 22, 24, 27, 20, 23, 25, 28, 29, 26, 31, 32, 35, 30, 33, 34, 37, 39, 38, 40, 41, 36, 42, 43, 45, 46, 48, 51, 49, 54, 57, 44, 47, 50, 52, 55, 58, 59, 53, 61, 62, 65, 60, 63, 64, 67, 69, 56, 68, 70, 71, 73, 74

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Oct 29 2001

The sequence would be a permutation of the naturals if Goldbach's conjecture holds (every even integer n greater than two is the sum of two primes). Inverse: A065307.

A065305, A065307.

Cf. A205666 (fixed points).

a065306 n = a065306_list !! (n-1)
a065306_list = map (subtract 2) $ f (concat a065305_tabl) [] where
   f (x:xs) ys = if x `elem` ys then f xs ys else x : f xs (x:ys)
-- Reinhard Zumkeller, Jan 30 2012

t[n_, k_] := (Prime[n] + Prime[k])/2; A065305 = Flatten[ Table[ t[n, k], {n, 2, 22}, {k, 2, n}]]; A065306 = (A065305 //. {a___, b_, c___, b_, d___} :> {a, b, c, d}) - 2 (* Jean-François Alcover, Jan 25 2012 *)

A065091 Odd primes.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Labos Elemer, Nov 12 2001

Rayes et al. prove that the a(n)-th Chebyshev-T polynomial, divided by x, is irreducible over the integers.
Odd primes can be written as a sum of no more than two consecutive positive integers. Powers of 2 do not have a representation as a sum of k consecutive positive integers (other than the trivial n=n, for k=1). See A111774. - Jaap Spies, Jan 04 2007
Intersection of A005408 and A000040. - Reinhard Zumkeller, Oct 14 2008
Primes which are the sum of two consecutive numbers. - Juri-Stepan Gerasimov, Nov 07 2009
The arithmetic mean of divisors of p^3, (1+p)(1+p^2)/4, for odd primes p is an integer. - Ctibor O. Zizka, Oct 20 2009
Primes == -+ 1 (mod 4). - Juri-Stepan Gerasimov, Apr 27 2010
a(n) = A053670(A179675(n)) and a(n) <> A053670(m) for m < A179675(n). - Reinhard Zumkeller, Jul 23 2010
Triads of the form <2*a(n+1), a(n+1), 3*a(n+1)> like <6,3,9>, <10,5,15>, <14,7,21> appear in the EKG sequence A064413, see Theorem (3) there. - Paul Curtz, Feb 13 2011.
Complement of A065090; abs(A151763(a(n))) = 1. - Reinhard Zumkeller, Oct 06 2011
Right edge of the triangle in A065305. - Reinhard Zumkeller, Jan 30 2012
Numbers with two odd divisors. - Omar E. Pol, Mar 24 2012
Odd prime p divides some (2^k + 1) or (2^k - 1), (k>0, minimal, cf. A003558) depending on the parity of A179480((p+1)/2) = r. This is a consequence of the Quasi-order theorem and corollaries, [Hilton and Pederson, pp. 260-264]: 2^k == (-1)^r mod b, b odd; and b divides 2^k - (-1)^r, where p is a subset of b. - Gary W. Adamson, Aug 26 2012
Subset of the arithmetic numbers (A003601). - Wesley Ivan Hurt, Sep 27 2013
Odd primes p satisfy the identity: p = (product(2*cos((2*k+1)*Pi/(2*p)), k=0..(p-3)/2))^2.  This follows from C(2*p, 0) = (-1)^((p-1)/2)*p, n>=2, with the minimal polynomial C(k,x) of rho(k) := 2*cos(Pi/k). See A187360 for C and the W. Lang link on the field Q(rho(n)), eqs. (20) and (37). - Wolfdieter Lang, Oct 23 2013
Numbers m > 1 such that m^2 divides (2m-1)!! + m. - Thomas Ordowski, Nov 28 2014
Numbers m such that m divides 2*(m-3)! + 1. - Thomas Ordowski, Jun 20 2015
Numbers m such that (2m-3)!! == m (mod m^2). - Thomas Ordowski, Jul 24 2016
Odd numbers m such that ((m-3)!!)^2 == +-1 (mod m). - Thomas Ordowski, Jul 27 2016
Primes of the form x^2 - y^2. - Thomas Ordowski, Feb 27 2017
Conjecture: a(n) is the smallest odd number m > prime(n) such that Sum_{k=1..prime(n)-1} k^(m-1) == prime(n)-1 (mod m). This is an extension of the Agoh-Giuga conjecture. - Thomas Ordowski, Aug 01 2018
Numbers k > 1 such that either Phi(k,x) == 1 (mod k) or Phi(k,x) == k (mod k^2) holds, where Phi(k,x) is the k-th cyclotomic polynomial. - Jianing Song, Aug 02 2018

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
M. O. Rayes, V. Trevisan and P. S. Wang, Factorization of Chebyshev Polynomials, ICM Report, 1998.
M. O. Rayes, V. Trevisan and P. S. Wang, Factorization of Chebyshev Polynomials, Computers & Mathematics with Applications, Volume 50, Issues 8-9, October-November 2005, Pages 1231-1240.
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A000040, A033270, union of A002144 and A002145.

Cf. A230953 (boustrophedon transform).

a065091 n = a065091_list !! (n-1)
a065091_list = tail a000040_list  -- Reinhard Zumkeller, Jan 30 2012

[NthPrime(n): n in [2..100]]; // Vincenzo Librandi, Jun 21 2015

A065091 := proc(n) RETURN(ithprime(n+1)) end:

Prime[Range[2, 33]] (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *)

forprime(p=3, 200, print1(p, ", ")) \\ Felix Fröhlich, Jun 30 2014

from sympy import prime
def A065091(n): return prime(n+1) # Chai Wah Wu, Jul 13 2024

def A065091_list(limit):  # after Minác's formula
    f = 3; P = [f]
    for n in range(3, limit, 2):
        if (f+1)>n*(f//n)+1: P.append(n)
        f = f*n
    return P
A065091_list(100)  # Peter Luschny, Oct 17 2013

a(n) = A000040(n+1). - M. F. Hasler, Oct 26 2013

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Jan 05 2002

Edited (moved contributions from A000040 to here) by M. F. Hasler, Oct 26 2013

A098090 Numbers k such that 2k-3 is prime.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 22, 23, 25, 28, 31, 32, 35, 37, 38, 41, 43, 46, 50, 52, 53, 55, 56, 58, 65, 67, 70, 71, 76, 77, 80, 83, 85, 88, 91, 92, 97, 98, 100, 101, 107, 113, 115, 116, 118, 121, 122, 127, 130, 133, 136, 137, 140, 142, 143, 148, 155, 157, 158

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 14 2004

Supersequence of A063908.

Left edge of the triangle in A065305. - Reinhard Zumkeller, Jan 30 2012

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A085090, A086801.
Numbers n such that 2n+k is prime: A005097 (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), this sequence (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).

Filtered([1..200], k-> IsPrime(2*k-3) ); # G. C. Greubel, May 21 2019

[ n: n in [1..200] | IsPrime(2*n-3) ]; // Vincenzo Librandi, Dec 26 2010

(Prime[Range[2,100]]+3)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[200],PrimeQ[2#-3]&] (* Harvey P. Dale, Mar 05 2022 *)

is(n)=isprime(2*n-3) \\ Charles R Greathouse IV, Feb 17 2017

[n for n in (1..200) if is_prime(2*n-3) ] # G. C. Greubel, May 21 2019

Half of p + 3, where p is a prime greater than 2.

A122845(a(n), 3) = 3; a(n) = A113935(n+1)/2. - Reinhard Zumkeller, Sep 14 2006

A116366 Triangle read by rows: even numbers as sums of two odd primes.

Original entry on oeis.org

6, 8, 10, 10, 12, 14, 14, 16, 18, 22, 16, 18, 20, 24, 26, 20, 22, 24, 28, 30, 34, 22, 24, 26, 30, 32, 36, 38, 26, 28, 30, 34, 36, 40, 42, 46, 32, 34, 36, 40, 42, 46, 48, 52, 58, 34, 36, 38, 42, 44, 48, 50, 54, 60, 62, 40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74, 44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82

Offset: 1

Reinhard Zumkeller, Feb 06 2006

T(n,k) = 2*A065305(n,k) = A065342(n+1,k+1);
Row sums give A116367; central terms give A116368;
T(n,1) = A113935(n+1);
T(n,n-2) = A048448(n) for n>2;
T(n,n-1) = A001043(n) for n>1;
T(n,n) = A001747(n+2) = A100484(n+1).

			Triangle begins:
  6;
  8,  10;
  10, 12, 14;
  14, 16, 18, 22;
  16, 18, 20, 24, 26;
  20, 22, 24, 28, 30, 34;
  22, 24, 26, 30, 32, 36, 38;
  26, 28, 30, 34, 36, 40, 42, 46;
  32, 34, 36, 40, 42, 46, 48, 52, 58;
  34, 36, 38, 42, 44, 48, 50, 54, 60, 62;
  40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74;
  44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82; etc. - _Bruno Berselli_, Aug 16 2013

G. C. Greubel, Rows n = 1..100 of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A001043, A001747, A048448, A065305, A065342, A100484, A113935, A116367, A116368.

[NthPrime(n+1)+NthPrime(k+1): k in [1..n], n in [1..15]]; // Bruno Berselli, Aug 16 2013

Table[Prime[n+1] + Prime[k+1], {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, May 12 2019 *)

{T(n,k) = prime(n+1) + prime(k+1)}; \\ G. C. Greubel, May 12 2019

[[nth_prime(n+1) + nth_prime(k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 12 2019

T(n,k) = prime(n+1) + prime(k+1), 1 <= k <= n.

A065342 Triangle of sum of two primes: prime(n)+prime(k) with n >= k >= 1.

Original entry on oeis.org

4, 5, 6, 7, 8, 10, 9, 10, 12, 14, 13, 14, 16, 18, 22, 15, 16, 18, 20, 24, 26, 19, 20, 22, 24, 28, 30, 34, 21, 22, 24, 26, 30, 32, 36, 38, 25, 26, 28, 30, 34, 36, 40, 42, 46, 31, 32, 34, 36, 40, 42, 46, 48, 52, 58, 33, 34, 36, 38, 42, 44, 48, 50, 54, 60, 62, 39, 40, 42, 44, 48

Offset: 1

Henry Bottomley, Oct 30 2001

			Sequence starts 2+2; 3+2, 3+3; 5+2, 5+3, 5+5; etc. i.e. 4; 5,6; 7,8,10; ...
Triangle begins:
   4;
   5,  6;
   7,  8, 10;
   9, 10, 12, 14;
  13, 14, 16, 18, 22;
  ...

Reinhard Zumkeller, First 125 rows of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A052147 (left edge), A100484 (right edge), A000040.
Cf. A087112.
Cf. A065305.

import Data.List (inits)
a065342 n k = a065342_tabl !! (n-1) !! (k-1)
a065342_row n = a065342_tabl !! (n-1)
a065342_tabl = zipWith (map . (+)) a000040_list $ tail $ inits a000040_list
-- Reinhard Zumkeller, Aug 02 2015, Jan 30 2012

row(n) = vector(n, k, prime(n)+prime(k)); \\ Michel Marcus, Sep 10 2021

T(n, k) = 2*A065305(n, k) [but note different offset].

A065307 "Inverse" to A065306.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13, 15, 14, 16, 17, 18, 19, 24, 20, 21, 25, 22, 26, 29, 23, 27, 28, 33, 30, 31, 34, 35, 32, 41, 36, 38, 37, 39, 40, 42, 43, 51, 44, 45, 52, 46, 48, 53, 47, 54, 58, 49, 55, 67, 50, 56, 57, 62, 59, 60, 63, 64, 61, 74, 65, 68, 66, 69, 70, 79

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Oct 29 2001

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

Cf. A065305, A065306.

Cf. A205666 (fixed points).

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a065307 n = (fromJust $ elemIndex n a065306_list) + 1
-- Reinhard Zumkeller, Jan 30 2012

A260580 Table read by rows: n-th row contains numbers not occurring earlier, that can be written as (p+q)/2 where p is the n-th odd prime, q <= p.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 11, 10, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 23, 24, 26, 29, 22, 25, 27, 30, 31, 28, 33, 34, 37, 32, 35, 36, 39, 41, 40, 42, 43, 38, 44, 45, 47, 48, 50, 53, 51, 56, 59, 46, 49, 52, 54, 57, 60, 61, 55, 63, 64, 67, 62, 65, 66, 69, 71

Offset: 1

Reinhard Zumkeller, Aug 11 2015

Length of n-th row = A105047(n+1);
T(n,1) = A260485(n);
T(n,A105047(n)) = A065091(n).

			Let p(n) = A065091(n) = prime(n+1):
.   n | p(n) | T(n,*)
. ----+------+----------------- ------------------------------------------
.   1 |    3 | [3]              3
.   2 |    5 | [4,5]            (5+3)/2,5
.   3 |    7 | [6,7]            (7+5)/2,7
.   4 |   11 | [8,9,11]         (11+5)/2,(11+7)/2,11
.   5 |   13 | [10,12,13]       (13+7)/2,(13+11)/2,13
.   6 |   17 | [14,15,17]       (17+11)/2,(17+13)/2,17
.   7 |   19 | [16,18,19]       (19+13)/2,(19+17)/2,19
.   8 |   23 | [20,21,23]       (23+17)/2,(23+19)/2,23
.   9 |   29 | [24,26,29]       (29+19)/2,(29+17)/2,29
.  10 |   31 | [22,25,27,30,31] (31+13)/2,(31+19)/2,(31+23)/2,(31+29)/2,31
.  11 |   37 | [28,33,34,37]    (37+19)/2,(37+29)/2,(37+31)/2,37
.  12 |   41 | [32,35,36,39,41] (41+23)/2,(41+29)/2,(41+31)/2,(41+37)/2,41

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A065305, A105047, A065091.

import Data.List.Ordered (union); import Data.List ((\\))
a260580 n k = a260580_tabf !! (n-1) !! (k-1)
a260580_row n = a260580_tabf !! (n-1)
a260580_tabf = zipWith (\\) (tail zss) zss where
                            zss = scanl union [] a065305_tabl

cp's OEIS Frontend

A065306 The Goldbach permutation: take A065305, cross out repetitions and subtract 2 from each term.

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A065091 Odd primes.

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A098090 Numbers k such that 2k-3 is prime.

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A116366 Triangle read by rows: even numbers as sums of two odd primes.

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A065342 Triangle of sum of two primes: prime(n)+prime(k) with n >= k >= 1.

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A065307 "Inverse" to A065306.

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A260580 Table read by rows: n-th row contains numbers not occurring earlier, that can be written as (p+q)/2 where p is the n-th odd prime, q <= p.

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