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%I #12 Feb 16 2025 08:33:26
%S 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,
%T 27,30,31,28,33,34,37,32,35,36,39,41,40,42,43,38,44,45,47,48,50,53,51,
%U 56,59,46,49,52,54,57,60,61,55,63,64,67,62,65,66,69,71
%N 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.
%C Length of n-th row = A105047(n+1);
%C T(n,1) = A260485(n);
%C T(n,A105047(n)) = A065091(n).
%H Reinhard Zumkeller, <a href="/A260580/b260580.txt">Rows n = 1..1000 of triangle, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e Let p(n) = A065091(n) = prime(n+1):
%e . n | p(n) | T(n,*)
%e . ----+------+----------------- ------------------------------------------
%e . 1 | 3 | [3] 3
%e . 2 | 5 | [4,5] (5+3)/2,5
%e . 3 | 7 | [6,7] (7+5)/2,7
%e . 4 | 11 | [8,9,11] (11+5)/2,(11+7)/2,11
%e . 5 | 13 | [10,12,13] (13+7)/2,(13+11)/2,13
%e . 6 | 17 | [14,15,17] (17+11)/2,(17+13)/2,17
%e . 7 | 19 | [16,18,19] (19+13)/2,(19+17)/2,19
%e . 8 | 23 | [20,21,23] (23+17)/2,(23+19)/2,23
%e . 9 | 29 | [24,26,29] (29+19)/2,(29+17)/2,29
%e . 10 | 31 | [22,25,27,30,31] (31+13)/2,(31+19)/2,(31+23)/2,(31+29)/2,31
%e . 11 | 37 | [28,33,34,37] (37+19)/2,(37+29)/2,(37+31)/2,37
%e . 12 | 41 | [32,35,36,39,41] (41+23)/2,(41+29)/2,(41+31)/2,(41+37)/2,41
%o (Haskell)
%o import Data.List.Ordered (union); import Data.List ((\\))
%o a260580 n k = a260580_tabf !! (n-1) !! (k-1)
%o a260580_row n = a260580_tabf !! (n-1)
%o a260580_tabf = zipWith (\\) (tail zss) zss where
%o zss = scanl union [] a065305_tabl
%Y Cf. A065305, A105047, A065091.
%K nonn,tabf,changed
%O 1,1
%A _Reinhard Zumkeller_, Aug 11 2015