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

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, 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, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3

Offset: 2

Author

N. J. A. Sloane, May 04 2012

Keywords

• nonn
• tabl
• nice

Comments

It is conjectured that pi(x) + pi(y) >= pi(x+y) for 1 < y <= x.

Examples

			Triangle begins:
  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, 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,
  ...
		

References

• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.

Links

• Reinhard Zumkeller, Rows n = 2..150 of triangle, flattened
• P. Erdős and J. L. Selfridge, Complete prime subsets of consecutive integers. 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).

Crossrefs

Cf. A000720, A212210, A212212, A212213, A060208, A047885, A047886.

Programs

Haskell

a212211 n k = a212211_tabl !! (n-2) !! (k-2)
a212211_tabl = map a212211_row [2..]
a212211_row n = zipWith (-)
   (map (+ a000720 n) $ take (n - 1) $ tail a000720_list)
   (drop (n + 1) a000720_list)
-- Reinhard Zumkeller, May 04 2012

Mathematica

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 *)