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A212210 Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 1, 1 <= k <= n, where pi() = A000720().

%I A212210 #34 Apr 27 2019 01:48:35
%S A212210 -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 A212210 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 A212210 -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 A212210 Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 1, 1 <= k <= n, where pi() = A000720().
%C A212210 It is conjectured that pi(x)+pi(y) >= pi(x+y) for 1 < y <= x.
%C A212210 A006093 gives row numbers of rows containing at least one negative term. [_Reinhard Zumkeller_, May 05 2012]
%D A212210 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.
%H A212210 Reinhard Zumkeller, <a href="/A212210/b212210.txt">Rows n = 1..150 of triangle, flattened</a>
%H A212210 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 A212210 Triangle begins:
%e A212210   -1
%e A212210   -1 0
%e A212210    0 0 1
%e A212210   -1 0 0 0
%e A212210    0 0 1 1 2
%e A212210   -1 0 1 1 1 1
%e A212210    0 1 2 1 2 1 2
%e A212210    0 1 1 1 1 1 2 2
%e A212210    0 0 1 0 1 1 2 1 1
%e A212210   -1 0 0 0 1 1 1 1 0 0
%e A212210   ...
%t A212210 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 A212210 (Haskell)
%o A212210 import Data.List (inits, tails)
%o A212210 a212210 n k = a212210_tabl !! (n-1) !! (k-1)
%o A212210 a212210_row n = a212210_tabl !! (n-1)
%o A212210 a212210_tabl = f $ tail $ zip (inits pis) (tails pis) where
%o A212210    f ((xs,ys) : zss) = (zipWith (-) (map (+ last xs) (xs)) ys) : f zss
%o A212210    pis = a000720_list
%o A212210 -- _Reinhard Zumkeller_, May 04 2012
%Y A212210 Cf. A000720, A212211, A212212, A212213, A060208, A047885, A047886.
%Y A212210 Left diagonal is -A010051.
%K A212210 sign,tabl,nice
%O A212210 1,15
%A A212210 _N. J. A. Sloane_, May 04 2012