A212212 Array read by antidiagonals: pi(n) + pi(k) - pi(n+k), where pi() = A000720.

-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, 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, 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

Offset: 1

Author

N. J. A. Sloane, May 04 2012

Keywords

• sign
• tabl
• nice

Comments

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

Examples

			Array begins:
  -1, -1,  0, -1,  0, -1,  0,  0,  0, -1,  0, -1, ...
  -1,  0,  0,  0,  0,  0,  1,  1,  0,  0,  0,  0, ...
   0,  0,  1,  0,  1,  1,  2,  1,  1,  0,  1,  1, ...
  -1,  0,  0,  0,  1,  1,  1,  1,  0,  0,  1,  1, ...
   0,  0,  1,  1,  2,  1,  2,  1,  1,  1,  2,  1, ...
  -1,  0,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   0,  1,  2,  1,  2,  1,  2,  2,  2,  1,  2,  1, ...
   0,  1,  1,  1,  1,  1,  2,  2,  1,  1,  1,  1, ...
   ...
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, 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-A212213, A060208, A047885, A047886. First row and column are -A010051.

Programs

Mathematica

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 *)