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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.

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