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

1,39

Comments

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

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).

Example

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, ...
   ...

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 *)

Crossrefs

Cf. A000720, A212210-A212213, A060208, A047885, A047886. First row and column are -A010051.

Sequence in context: A047885 A072731 A221169 * A212213 A214339 A129174

Adjacent sequences: A212209 A212210 A212211 * A212213 A212214 A212215

Keyword

sign,tabl,nice

Author

N. J. A. Sloane, May 04 2012

Status

approved