%I #11 Jul 19 2024 15:21:38
%S 1,2,1,2,3,4,3,4,3,6,3,6,3,6,5,6,3,8,3,8,5,6,3,10,5,6,5,10,3,10,3,8,5,
%T 6,7,14,3,6,5,12,3,12,3,10,11,6,3,14,5,10,5,10,3,12,9,12,5,6,3,18,3,6,
%U 11,10,9,14,3,10,5,16,3,18,3,6,9,10,7,14,3
%N Number of distinct differences in row n of the reciprocity array of 1.
%C The "reciprocity law" that Sum{[(n*k+x)/m] : k = 0..m} = Sum{[(m*k+x)/n] : k = 0..n} where x is a real number and m and n are positive integers, is proved in Section 3.5 of Concrete Mathematics (see References). See A259572 for a guide to related sequences.
%D R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%H Antti Karttunen, <a href="/A259576/b259576.txt">Table of n, a(n) for n = 1..5000</a>
%e In the array at A259575, row 6 is (1,3,6,8,11,15,16,18,...), with differences (2,3,2,3,4,1,2,...), and distinct differences {1,2,3,4}, so that a(4) = 4.
%t x = 1; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
%t t[m_] := Table[s[m, n], {n, 1, 1000}];
%t u = Table[Length[Union[Differences[t[m]]]], {m, 1, 120}] (* A259576 *)
%o (PARI)
%o A259575sq(m,n) = sum(k=0,m-1,(1+(n*k))\m);
%o A259576(n) = #Set(vector(n,k,A259575sq(n,1+k)-A259575sq(n,k))); \\ _Antti Karttunen_, Mar 02 2023
%Y Cf. A249572, A249573, A259575.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jul 01 2015