OFFSET
1,2
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.
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.
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).
EXAMPLE
Triangle starts:
1;
3, 4;
15, 10, 6;
105, 70, 126, 120;
1155, 1540, 1386, 330, 210;
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 *)
PROG
(PARI) T(n, k) = {my(val = prod(j=1, n, prime(j))/prime(k)); val * lift(1/Mod(val, prime(k))); }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Michel Marcus, Apr 10 2014
STATUS
approved