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%I #5 Dec 29 2023 12:52:12
%S 1,1,3,1,4,5,1,6,7,8,1,7,8,9,11,1,9,10,11,13,14,1,10,11,12,14,15,17,1,
%T 12,13,14,16,17,19,20,1,15,16,17,19,20,22,23,25,1,16,17,18,20,21,23,
%U 24,26,29,1,19,20,21,23,24,26,27,29,32,33,1,21,22,23,25,26,28,29,31,34,35
%N Triangle read by rows: a(n,m) = If(n = 1, then 1, else Prime(n) - 1 + Sum_{k=n..m} (Prime(k + 1) - Prime(k))/2 ).
%C An improved triangular Goldbach sequence in which the gap sum is taken from a start at n.
%e 1
%e 1, 3
%e 1, 4, 5
%e 1, 6, 7, 8
%e 1, 7, 8, 9, 11
%e 1, 9, 10, 11, 13, 14
%e 1, 10, 11, 12, 14, 15, 17
%e 1, 12, 13, 14, 16, 17, 19, 20
%e 1, 15, 16, 17, 19, 20, 22, 23, 25
%e 1, 16, 17, 18, 20, 21, 23, 24, 26, 29
%t t[n_, m_] := If[n == 1, 1, Prime[n] + Sum[(Prime[k + 1] - Prime[k])/2, {k, n, m}] - 1]; Table[ t[n, m], {m, 11}, {n, m}] // Flatten
%Y Main diagonal: A078444, 2nd diagonal: A073273.
%Y Columns 1-8: A000012, A006254, A098090, A089253, A097069, A097338, A097480, A098605.
%K nonn,tabl
%O 1,3
%A _Roger L. Bagula_, May 04 2006
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