A278716 Triangle read by rows: T(k, h) gives the denominators of the Dedekind sums s(h, k) with gcd(h, k) = 1 or 0 otherwise.
1, 18, 18, 8, 1, 8, 5, 1, 1, 5, 18, 1, 1, 1, 18, 14, 14, 14, 14, 14, 14, 16, 1, 16, 1, 16, 1, 16, 27, 27, 1, 27, 27, 1, 27, 27, 5, 1, 1, 1, 1, 1, 1, 1, 5, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 72, 1, 1, 1, 72, 1, 72, 1, 1, 1, 72, 13, 13, 13, 13, 1, 13, 13, 1, 13, 13, 13, 13
Offset: 2
Examples
The triangle T(k, h) begins (here l is used if gcd(h, k) > 1 instead of 1): k\h 1 2 3 4 5 6 7 8 9 10 11 12 2: 1 3: 18 18 4: 8 l 8 5: 5 1 1 5 6: 18 l l l 18 7: 14 14 14 14 14 14 8: 16 l 16 l 16 l 16 9: 27 27 l 27 27 l 27 27 10: 5 l 1 l l l 1 l 5 11: 22 22 22 22 22 22 22 22 22 22 12: 72 l l l 72 l 72 l l l 72 13: 13 13 13 13 1 13 13 1 13 13 13 13 ... n = 14: 14 l 14 l 14 l l l 14 l 14 l 14, n = 15: 90 18 l 90 l l 18 18 l l 90 l 18 90. ...
Links
- G. C. Greubel, Rows n=2..100 of triangle, flattened
Crossrefs
Cf. A278715.
Programs
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Magma
[[GCD(n,k) eq 1 select Denominator((&+[(j/n)*(k*j/n - Floor(k*j/n) - 1/2): j in [1..(n-1)]])) else 0: k in [1..(n-1)]]: n in [2..10]]; // G. C. Greubel, Nov 22 2018
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Mathematica
T[n_, k_]:= If[GCD[n, k] == 1, Sum[(j/n)*(k*j/n - Floor[k*j/n] - 1/2), {j, 1, n - 1}], 0]; Denominator[Table[T[n, k], {n, 2, 15}, {k, 1, n - 1}]]//Flatten (* G. C. Greubel, Nov 22 2018 *) -
PARI
{T(n,k) = if(gcd(n,k)==1, sum(j=1,n-1, (j/n)*(k*j/n - floor(k*j/n) - 1/2)), 0)}; for(n=2,15, for(k=1,n-1, print1(denominator(T(n,k)), ", "))) \\ G. C. Greubel, Nov 22 2018 -
Sage
def T(n,k): if gcd(n,k)==1: return denominator(sum((j/n)*(k*j/n - floor(k*j/n) - 1/2) for j in (1..(n-1)))) elif gcd(n,k)!=1: return 0 else: 0 [[T(n,k) for k in (1..n-1)] for n in (2..15)] # G. C. Greubel, Nov 22 2018
Formula
T(k ,h) = denominator(s(h,k)) with the Dedekind sums s(h,k) given in a comment on A278715 and gcd(h,k) = 1. k >= 2, h = 1, 2, ..., k-1. If gcd(h,k) > 1 then T(h, k) = 1 (from s(h,k) put to 0).
Comments