A260580 - OEIS

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.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Length of n-th row = A105047(n+1);

T(n,1) = A260485(n);

T(n,A105047(n)) = A065091(n).

LINKS

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

EXAMPLE

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

MATHEMATICA

n = 4; p = 3; pending = {}; leftover = {}; ans = {}; While[n < 142, n = n + 2; nop = Length[pending]; If[nop > 0, Do[m = pending[[j]]; If[PrimeQ[m - p], AppendTo[ans, m/2], AppendTo[leftover, m]], {j, 1, nop}]; pending = leftover; leftover = {}]; If[PrimeQ[n - p], AppendTo[ans, n/2], AppendTo[pending, n]]; If[n == 2*p, p = NextPrime[p]]]; Print[ans] (* Lei Zhou, Mar 05 2026 *)

PROG

(Haskell)

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

CROSSREFS

Cf. A065305, A105047, A065091.

Sequence in context: A138220 A137913 A137937 * A297468 A047565 A026466

Adjacent sequences: A260577 A260578 A260579 * A260581 A260582 A260583

KEYWORD

nonn,tabf

AUTHOR

Reinhard Zumkeller, Aug 11 2015

STATUS

approved