cp's OEIS Frontend

A212211 Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 2, 2 <= k <= n, where pi() = A000720().

%I A212211 #30 Sep 07 2023 04:42:40
%S A212211 0,0,1,0,0,0,0,1,1,2,0,1,1,1,1,1,2,1,2,1,2,1,1,1,1,1,2,2,0,1,0,1,1,2,
%T A212211 1,1,0,0,0,1,1,1,1,0,0,0,1,1,2,1,2,1,1,1,2,0,1,1,1,1,1,1,1,1,1,1,1,2,
%U A212211 1,2,1,2,2,2,1,2,2,3
%N A212211 Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 2, 2 <= k <= n, where pi() = A000720().
%C A212211 It is conjectured that pi(x) + pi(y) >= pi(x+y) for 1 < y <= x.
%D A212211 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.
%H A212211 Reinhard Zumkeller, <a href="/A212211/b212211.txt">Rows n = 2..150 of triangle, flattened</a>
%H A212211 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 A212211 Triangle begins:
%e A212211   0,
%e A212211   0, 1,
%e A212211   0, 0, 0,
%e A212211   0, 1, 1, 2,
%e A212211   0, 1, 1, 1, 1,
%e A212211   1, 2, 1, 2, 1, 2,
%e A212211   1, 1, 1, 1, 1, 2, 2,
%e A212211   0, 1, 0, 1, 1, 2, 1, 1,
%e A212211   0, 0, 0, 1, 1, 1, 1, 0, 0,
%e A212211   0, 1, 1, 2, 1, 2, 1, 1, 1, 2,
%e A212211   0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
%e A212211   ...
%t A212211 t[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n+k]; Flatten[ Table[t[n, k], {n, 2, 13}, {k, 2, n}]] (* _Jean-François Alcover_, May 21 2012 *)
%o A212211 (Haskell)
%o A212211 a212211 n k = a212211_tabl !! (n-2) !! (k-2)
%o A212211 a212211_tabl = map a212211_row [2..]
%o A212211 a212211_row n = zipWith (-)
%o A212211    (map (+ a000720 n) $ take (n - 1) $ tail a000720_list)
%o A212211    (drop (n + 1) a000720_list)
%o A212211 -- _Reinhard Zumkeller_, May 04 2012
%Y A212211 Cf. A000720, A212210, A212212, A212213, A060208, A047885, A047886.
%K A212211 nonn,tabl,nice
%O A212211 2,10
%A A212211 _N. J. A. Sloane_, May 04 2012