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%I #30 Sep 07 2023 04:27:41
%S -1,-1,-1,0,0,0,-1,0,0,-1,0,0,1,0,0,-1,0,0,0,0,-1,0,0,1,0,1,0,0,0,1,1,
%T 1,1,1,1,0,0,1,2,1,2,1,2,1,0,-1,0,1,1,1,1,1,1,0,-1,0,0,1,1,2,1,2,1,1,
%U 0,0,-1,0,0,0,1,1,1,1,0,0,0,-1,0,0,1,0,1,1,2,1,1,0,1,0,0,0,1,1,1,1,1,2,2,1,1,1,1,1,0
%N Array read by antidiagonals: pi(n) + pi(k) - pi(n+k), where pi() = A000720.
%C It is conjectured that pi(x) + pi(y) >= pi(x+y) for 1 < y <= x.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.
%H G. C. Greubel, <a href="/A212212/b212212.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H P. Erdős 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 Array begins:
%e -1, -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, ...
%e -1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, ...
%e 0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, ...
%e -1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, ...
%e 0, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, ...
%e -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, ...
%e 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, ...
%e ...
%t a[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n+k]; Flatten[ Table[a[n-k, k], {n, 1, 15}, {k, 1, n-1}]] (* _Jean-François Alcover_, Jul 18 2012 *)
%Y Cf. A000720, A212210-A212213, A060208, A047885, A047886. First row and column are -A010051.
%K sign,tabl,nice
%O 1,39
%A _N. J. A. Sloane_, May 04 2012
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