A345647 Square array read by downward antidiagonals: A(n, k) = number of primes in the interval [n+1, n+k], n >= 1, k > 1.

1, 1, 0, 2, 1, 1, 2, 1, 1, 0, 3, 2, 2, 1, 1, 3, 2, 2, 1, 1, 0, 3, 2, 2, 1, 1, 0, 0, 3, 2, 2, 1, 1, 0, 0, 0, 4, 3, 3, 2, 2, 1, 1, 1, 1, 4, 3, 3, 2, 2, 1, 1, 1, 1, 0, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 0, 5, 4, 4, 3, 3, 2, 2, 2, 2

Offset

1,4

Comments

Conjecture: If A(n, k) = 0, then A001221(Product_{x=1..k}(n+x)) >= k (cf. Wikipedia).

Links

Table of n, a(n) for n=1..87.

Wikipedia, Grimm's conjecture

Example

The array starts as follows:
  1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6
  0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6
  1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6
  0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5
  1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5
  0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5
  0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5
  0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5
  1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5
  0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4
  1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4
  0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4
  0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4
  0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5
  1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5
  0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4

Prog

(PARI) a(n, k) = primepi(n+k)-primepi(n+1)
array(rows, cols) = for(x=1, rows, for(y=2, cols+1, print1(a(x, y), ", ")); print(""))
array(16, 16) \\ Print initial 16 rows and 16 columns of the array

Crossrefs

Cf. A001221, A101083.

Sequence in context: A308424 A317489 A391280 * A091950 A014750 A015491

Adjacent sequences: A345644 A345645 A345646 * A345648 A345649 A345650

Keyword

nonn,tabl

Author

Felix Fröhlich, Jun 21 2021

Status

approved