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

%I A248623 #42 Sep 23 2023 03:39:47
%S A248623 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,
%T A248623 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,
%U A248623 4,2,3,3,1,4,2,2,2,2
%N A248623 Triangle T(n,k) read by rows, of number of primes in interval [k*n, (k+1)*n] exclusive, n>=k.
%C A248623 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.
%C A248623 First column (k=1) is A060715.
%C A248623 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).
%C A248623 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].
%C A248623 The falling diagonal starting with T(2,2) (i.e., the sequence when n=k>=2) is A089610(n).
%C A248623 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).
%H A248623 M. El Bachraoui, <a href="http://www.m-hikari.com/ijcms-password/ijcms-password13-16-2006/elbachraouiIJCMS13-16-2006.pdf">Primes in the interval [2n,3n]</a>, Int. J. Contemp. Math. Sciences 1:13 (2006), pp. 617-621.
%H A248623 A. Loo, <a href="http://www.m-hikari.com/ijcms-2011/37-40-2011/looIJCMS37-40-2011.pdf">On the primes in the interval [3n,4n]</a>, Int. J. Contemp. Math. Sciences 6 (2011), no. 38, 1871-1882.
%H A248623 S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram24.html">A proof of Bertrand's postulate</a>, J. Indian Math. Soc., 11 (1919), 181-182.
%H A248623 Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.html">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. <a href="http://arxiv.org/abs/1212.2785">arXiv:1212.2785</a>
%H A248623 Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann%27s_conjecture">Oppermann's conjecture</a>
%e A248623 Triangle starts:
%e A248623   0
%e A248623   1  1
%e A248623   1  1  1
%e A248623   2  1  1  2
%e A248623   1  2  2  1  1
%e A248623   2  2  2  1  1  2
%e A248623   2  2  1  2  2  2  1
%e A248623   2  3  2  1  2  1  2  2
%e A248623   3  2  2  3  2  2  2  2  2
%e A248623   4  2  2  3  2  2  3  2  1  4
%e A248623   3  3  3  2  2  3  2  2  4  1  2
%e A248623   4  2  4  2  3  3  1  4  2  2  2  2
%e A248623 T(1,1) = 0.
%e A248623 T(11,9) = 4 because the number of primes in [99,110] is 4: {101, 103, 107, 109}.
%t A248623 T[n_, k_] := PrimePi[(k+1)n] - PrimePi[n k] - Boole[PrimeQ[(k+1)n]];
%t A248623 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 23 2018, from PARI *)
%o A248623 (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
%Y A248623 Cf. A060715, A002620, A218831, A089610 (related).
%K A248623 nonn,tabl
%O A248623 1,7
%A A248623 _Bob Selcoe_, Oct 10 2014