The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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
 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 (list; table; graph; refs; listen; history; text; internal format)
 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). T(n, n) = A002110(n-1) * A079276(n). - Peter Luschny, Apr 12 2014 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 Cf. A000040 (primes), A002110 (primorials), A070826 (first column), A079276. Sequence in context: A338438 A130113 A004735 * A066830 A192211 A083061 Adjacent sequences: A240670 A240671 A240672 * A240674 A240675 A240676 KEYWORD nonn,tabl AUTHOR Michel Marcus, Apr 10 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 20:21 EST 2022. Contains 358588 sequences. (Running on oeis4.)