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.

1, 3, 4, 15, 10, 6, 105, 70, 126, 120, 1155, 1540, 1386, 330, 210, 15015, 20020, 6006, 25740, 16380, 6930, 255255, 170170, 306306, 145860, 46410, 157080, 450450, 4849845, 3233230, 3879876, 8314020, 6172530, 3730650, 9129120, 9189180

Offset: 1

Author

Michel Marcus, Apr 10 2014

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle starts:
     1;
     3,    4;
    15,   10,    6;
   105,   70,  126,  120;
  1155, 1540, 1386,  330,  210;
		

Links

• Matthias Schmitt, A function to calculate all relative prime numbers up to the product of the first n primes, arXiv:1404.0706 [math.NT] (see Example table on page 9).
• Ramin Zahedi, On algebraic structure of the set of prime numbers, arXiv:1209.3165 [math.GM], 2012.
• Ramin Zahedi, 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, arXiv:1408.1888 [math.GM], 2014.

Crossrefs

Cf. A000040 (primes), A002110 (primorials), A070826 (first column), A079276.

Programs

Maple

T := proc(n, k)
v := mul(ithprime(j), j=1..n)/ithprime(k);
v * ((1/v) mod ithprime(k)) end:
seq(print(seq(T(n,k), k=1..n)), n=1..7); # Peter Luschny, Apr 12 2014

Mathematica

lift[Rational[1, n_], p_] := Module[{m}, m /. Solve[n*m == 1, m, Modulus -> p][[1]]]; lift[1, 2] = 1;
v[n_, k_] := Product[Prime[j], {j, 1, n}]/Prime[k];
T[n_, k_] := v[n, k]*lift[1/v[n, k], Prime[k]];
Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 30 2017 *)

PARI

T(n, k) = {my(val = prod(j=1, n, prime(j))/prime(k)); val * lift(1/Mod(val, prime(k)));}

Formula

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).

T(n, n) = A002110(n-1) * A079276(n). - Peter Luschny, Apr 12 2014