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%I #34 Apr 27 2019 01:48:35
%S -1,-1,0,0,0,1,-1,0,0,0,0,0,1,1,2,-1,0,1,1,1,1,0,1,2,1,2,1,2,0,1,1,1,
%T 1,1,2,2,0,0,1,0,1,1,2,1,1,-1,0,0,0,1,1,1,1,0,0,0,0,1,1,2,1,2,1,1,1,2,
%U -1,0,1,1,1,1,1,1,1,1,1,1,0,1,2,1,2,1,2,2,2,1,2,2,3,0,1,1,1,1,1,2,2,1,1,2,2,3,3
%N Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 1, 1 <= k <= n, where pi() = A000720().
%C It is conjectured that pi(x)+pi(y) >= pi(x+y) for 1 < y <= x.
%C A006093 gives row numbers of rows containing at least one negative term. [_Reinhard Zumkeller_, May 05 2012]
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.
%H Reinhard Zumkeller, <a href="/A212210/b212210.txt">Rows n = 1..150 of triangle, flattened</a>
%H P. Erdos and J. L. Selfridge, <a href="http://www.renyi.hu/~p_erdos/1971-03.pdf">Complete prime subsets of consecutive integers</a>. Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971), pp. 1--14. Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971. MR0337828 (49 #2597).
%e Triangle begins:
%e -1
%e -1 0
%e 0 0 1
%e -1 0 0 0
%e 0 0 1 1 2
%e -1 0 1 1 1 1
%e 0 1 2 1 2 1 2
%e 0 1 1 1 1 1 2 2
%e 0 0 1 0 1 1 2 1 1
%e -1 0 0 0 1 1 1 1 0 0
%e ...
%t t[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n + k]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 17 2012 *)
%o (Haskell)
%o import Data.List (inits, tails)
%o a212210 n k = a212210_tabl !! (n-1) !! (k-1)
%o a212210_row n = a212210_tabl !! (n-1)
%o a212210_tabl = f $ tail $ zip (inits pis) (tails pis) where
%o f ((xs,ys) : zss) = (zipWith (-) (map (+ last xs) (xs)) ys) : f zss
%o pis = a000720_list
%o -- _Reinhard Zumkeller_, May 04 2012
%Y Cf. A000720, A212211, A212212, A212213, A060208, A047885, A047886.
%Y Left diagonal is -A010051.
%K sign,tabl,nice
%O 1,15
%A _N. J. A. Sloane_, May 04 2012
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