A278716 - OEIS

%I #20 Sep 08 2022 08:46:18

%S 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,

%T 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,

%U 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

%N 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.

%C For the numerators see A278715, also for references and details of the Dedekind sums s(h, k).

%H G. C. Greubel, <a href="/A278716/b278716.txt">Rows n=2..100 of triangle, flattened</a>

%F 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).

%e The triangle T(k, h) begins (here l is used if gcd(h, k) > 1 instead of 1):

%e k\h  1  2  3  4   5  6   7   8  9 10 11 12

%e 2:   1

%e 3:  18 18

%e 4:   8  l  8

%e 5:   5  1  1  5

%e 6:  18  l  l  l  18

%e 7:  14 14 14 14  14  14

%e 8:  16  l 16  l  16  l  16

%e 9:  27 27  l 27  27  l  27  27

%e 10:  5  l  1  l   l  l   1   l  5

%e 11: 22 22 22 22  22 22  22  22 22 22

%e 12: 72  l  l  l  72  l  72   l  l  l 72

%e 13: 13 13 13 13   1 13  13   1 13 13 13 13

%e ...

%e n = 14: 14 l 14 l 14 l l l 14 l 14 l 14,

%e n = 15: 90 18 l 90 l l 18 18 l l 90 l 18 90.

%e ...

%t 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 *)

%o (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)};

%o for(n=2,15, for(k=1,n-1, print1(denominator(T(n,k)), ", "))) \\ _G. C. Greubel_, Nov 22 2018

%o (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

%o (Sage)

%o def T(n,k):

%o     if gcd(n,k)==1:

%o        return denominator(sum((j/n)*(k*j/n - floor(k*j/n) - 1/2) for j in (1..(n-1))))

%o     elif gcd(n,k)!=1:

%o         return 0

%o     else:

%o         0

%o [[T(n,k) for k in (1..n-1)] for n in (2..15)] # _G. C. Greubel_, Nov 22 2018

%Y Cf. A278715.

%K nonn,tabl,frac,easy

%O 2,2

%A _Wolfdieter Lang_, Nov 28 2016