A319039 Triangle read by rows: T(n,k), n >= 1, k = 0..A005867(n), is the smallest integer m > 0 such that the interval [P(n)*m+1, P(n)*(m+1)] includes exactly k primes, where P(n) = A002110(n) is the n-th primorial, or 0 if no such m exists.

4, 1, 15, 4, 1, 360, 83, 17, 26, 10, 4, 3, 1, 0, 1751793, 235449, 60110, 10471, 17110, 8495, 6288, 3182, 2452, 1349, 331, 348, 446, 223, 249, 205, 111, 67, 55, 63, 28, 37, 14, 21, 18, 11, 10, 6, 551, 5, 4, 7, 3, 2

Offset: 1

Author

Jon E. Schoenfield, Dec 13 2018

Keywords

• nonn
• tabf
• more

Comments

For any interval I of length P(n) that starts beyond prime(n), divisibility by one or more of the first n primes limits the maximum number of primes in I to Product_{j=1..n} (prime(j) - 1) = A005867(n). Beyond this, for n >= 3, divisibility by primes larger than prime(n) (but smaller than P(n)) ensures that T(n,k)=0 for one or more terms at the end of the row. E.g., for every value of m mod 7, at least one of the A005867(3) = 8 numbers in [30*m+1, 30*(m+1)] that is not divisible by 2, 3, or 5 -- i.e., at least one of the 8 numbers {M+1, M+7, M+11, M+13, M+17, M+19, M+23, M+29} where M=30*m -- will be divisible by 7, so T(3,8)=0. A similar argument shows that T(4,46) = T(4,47) = T(4,48) = 0.

Examples

			Table begins
n=1: [4, 1];
n=2: [15, 4, 1];
n=3: [360, 83, 17, 26, 10, 4, 3, 1, 0];
n=4: [1751793, 235449, 60110, 10471, 17110, 8495, 6288, 3182, 2452, 1349, 331, 348, 446, 223, 249, 205, 111, 67, 55, 63, 28, 37, 14, 21, 18, 11, 10, 6, 551, 5, 4, 7, 3, 2, ?, 1, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, 0, 0, 0];
		

Crossrefs

Cf. A002110, A005867.