A212213 Array read by antidiagonals: pi(n) + pi(k) - pi(n+k), where pi() = A000720.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 1, 2, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset

2,23

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.

Example

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

Mathematica

t[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n + k]; Table[t[n - k + 2, k], {n, 0, 15}, {k, 2, n}] // Flatten (* Jean-François Alcover, Dec 31 2012 *)

Crossrefs

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

Sequence in context: A072731 A221169 A212212 * A214339 A129174 A129175

Adjacent sequences: A212210 A212211 A212212 * A212214 A212215 A212216

Keyword

nonn,tabl,nice

Author

N. J. A. Sloane, May 04 2012

Status

approved