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A240673
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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.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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Triangle starts:
1;
3, 4;
15, 10, 6;
105, 70, 126, 120;
1155, 1540, 1386, 330, 210;
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MAPLE
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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
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MATHEMATICA
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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]];
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PROG
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(PARI) T(n, k) = {my(val = prod(j=1, n, prime(j))/prime(k)); val * lift(1/Mod(val, prime(k))); }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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