This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A240673 #41 Oct 13 2019 11:19:08
%S A240673 1,3,4,15,10,6,105,70,126,120,1155,1540,1386,330,210,15015,20020,6006,
%T A240673 25740,16380,6930,255255,170170,306306,145860,46410,157080,450450,
%U A240673 4849845,3233230,3879876,8314020,6172530,3730650,9129120,9189180
%N A240673 Triangle read by rows: T(n, k) = v(n, k)*((1/v(n, k)) mod prime(k)), where v(n, k) = (Product_{j=1..n} prime(j))/prime(k), n >= 1, 1 <= k <= n.
%C A240673 These coefficients have the following property: for any number j in 0..primorial(n)-1, j = Sum_{i=1..n} T(n,k)*(j mod prime(i)) mod primorial(n). For example, with the first 3 primes (2, 3, and 5) and j=47, j is [47 mod 2, 47 mod 3, 47 mod 5] = [1, 2, 2], and 15*1 + 10*2 + 6*2 = 15 + 20 + 12 = 47.
%H A240673 Matthias Schmitt, <a href="http://arxiv.org/abs/1404.0706">A function to calculate all relative prime numbers up to the product of the first n primes</a>, arXiv:1404.0706 [math.NT] (see Example table on page 9).
%H A240673 Ramin Zahedi, <a href="http://arxiv.org/abs/1209.3165">On algebraic structure of the set of prime numbers</a>, arXiv:1209.3165 [math.GM], 2012.
%H A240673 Ramin Zahedi, <a href="http://arxiv.org/abs/1408.1888">On a Deterministic Property of the Category of k-th Numbers: A Deterministic Structure Based on a Linear Function for Redefining the k-th Numbers in Certain Intervals</a>, arXiv:1408.1888 [math.GM], 2014.
%F A240673 T(n, k) = v(n, k) * ((1/v(n, k)) mod prime(k)), where v(n, k) = Product_{j=1..n} prime(j)/prime(k).
%F A240673 T(n, n) = A002110(n-1) * A079276(n). - _Peter Luschny_, Apr 12 2014
%e A240673 Triangle starts:
%e A240673 1;
%e A240673 3, 4;
%e A240673 15, 10, 6;
%e A240673 105, 70, 126, 120;
%e A240673 1155, 1540, 1386, 330, 210;
%p A240673 T := proc(n, k)
%p A240673 v := mul(ithprime(j), j=1..n)/ithprime(k);
%p A240673 v * ((1/v) mod ithprime(k)) end:
%p A240673 seq(print(seq(T(n,k), k=1..n)), n=1..7); # _Peter Luschny_, Apr 12 2014
%t A240673 lift[Rational[1, n_], p_] := Module[{m}, m /. Solve[n*m == 1, m, Modulus -> p][[1]]]; lift[1, 2] = 1;
%t A240673 v[n_, k_] := Product[Prime[j], {j, 1, n}]/Prime[k];
%t A240673 T[n_, k_] := v[n, k]*lift[1/v[n, k], Prime[k]];
%t A240673 Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 30 2017 *)
%o A240673 (PARI) T(n, k) = {my(val = prod(j=1, n, prime(j))/prime(k)); val * lift(1/Mod(val, prime(k)));}
%Y A240673 Cf. A000040 (primes), A002110 (primorials), A070826 (first column), A079276.
%K A240673 nonn,tabl
%O A240673 1,2
%A A240673 _Michel Marcus_, Apr 10 2014