A248623 Triangle T(n,k) read by rows, of number of primes in interval [k*n, (k+1)*n] exclusive, n>=k.

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

Offset: 1

Author

Bob Selcoe, Oct 10 2014

Keywords

• nonn
• tabl

Comments

Reading by columns: First column (k=1) is the number of primes in the interval [n,2n], n>=1; second column is the number of primes in the interval [2n,3n], n>=2; third column is the number of primes in the interval [3n,4n], n>=3; etc.

First column (k=1) is A060715.

Proofs exist which state that for n>1, at least one prime is in [n,2n] ("Bertrand's Postulate", first proved by P. Chebyshev), [2n,3n] (proved by El Bachraoui) and [3n,4n] (proved by Loo).

Starting with T(2,1), the falling diagonal of the first 2 numbers in each column (read by column) are the number of primes in [A002620(n), A002620(n+1)], n>=3. That is, the coefficients of T(2,1), T(3,1), T(3,2), T(4,2), T(4,3), T(5,3) etc. are the number of primes between A002620(n) and A002620(n+1), n>=3. This pertains to Oppermann's conjecture, which states there is at least one prime in [n^2, n^2+n] and [n^2+n, (n+1)^2].

The falling diagonal starting with T(2,2) (i.e., the sequence when n=k>=2) is A089610(n).

Except for trivial T(1,1) = 0 (null interval [1,2]) it is conjectured here that at least one prime is in [k*n, (k+1)*n] exclusive, n>=k. That is, all the coefficients in the triangle are positive, except T(1,1).

Examples

			Triangle starts:
  0
  1  1
  1  1  1
  2  1  1  2
  1  2  2  1  1
  2  2  2  1  1  2
  2  2  1  2  2  2  1
  2  3  2  1  2  1  2  2
  3  2  2  3  2  2  2  2  2
  4  2  2  3  2  2  3  2  1  4
  3  3  3  2  2  3  2  2  4  1  2
  4  2  4  2  3  3  1  4  2  2  2  2
T(1,1) = 0.
T(11,9) = 4 because the number of primes in [99,110] is 4: {101, 103, 107, 109}.
		

Links

• M. El Bachraoui, Primes in the interval [2n,3n], Int. J. Contemp. Math. Sciences 1:13 (2006), pp. 617-621.
• A. Loo, On the primes in the interval [3n,4n], Int. J. Contemp. Math. Sciences 6 (2011), no. 38, 1871-1882.
• S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
• Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785
• Wikipedia, Oppermann's conjecture

Crossrefs

Cf. A060715, A002620, A218831, A089610 (related).

Programs

Mathematica

T[n_, k_] := PrimePi[(k+1)n] - PrimePi[n k] - Boole[PrimeQ[(k+1)n]];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 23 2018, from PARI *)

PARI

tabl(nn) = {for (n=1, nn, for (k=1, n, print1(primepi((k+1)*n) - primepi(n*k) - isprime((k+1)*n), ", ");); print(););} \\ Michel Marcus, Nov 04 2014