OFFSET
1,7
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).
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
EXAMPLE
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}.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Bob Selcoe, Oct 10 2014
STATUS
approved