A259573 Number of distinct differences in row n of the reciprocity array of 0.

1, 2, 3, 4, 3, 4, 3, 6, 5, 6, 3, 8, 3, 6, 7, 8, 3, 8, 3, 8, 9, 6, 3, 12, 5, 6, 7, 10, 3, 14, 3, 10, 9, 6, 9, 14, 3, 6, 9, 12, 3, 12, 3, 12, 11, 6, 3, 18, 5, 10, 9, 12, 3, 12, 9, 14, 9, 6, 3, 22, 3, 6, 13, 12, 9, 14, 3, 12, 9, 14, 3, 18, 3, 6, 13, 12, 9, 16

Offset

1,2

Comments

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).

References

R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Example

In the array at A259572, row 4 is (0,2,3,6,6,8,9,12,12,14,15,...), with differences (2,1,3,0,2,1,3,0,2,1,3,0, ...), and distinct differences {0,1,2,3}, so that a(4) = 4. Example corrected by Antti Karttunen, Nov 30 2021

Mathematica

x = 0;  s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
t[m_] := Table[s[m, n], {n, 1, 1000}];
u = Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]  (* A259573 *)

Prog

(PARI)
A259572(m, n) = ((m*n - m - n + gcd(m, n))/2); \\ After Witold Dlugosz's formula for A259572.
A259573(n) = #Set(vector(n, k, A259572(n, 1+k)-A259572(n, k))); \\ Antti Karttunen, Nov 30 2021

Crossrefs

Cf. A249572, A259574.

Sequence in context: A304730 A323374 A324197 * A108015 A344322 A030398

Adjacent sequences: A259570 A259571 A259572 * A259574 A259575 A259576

Keyword

nonn,easy

Author

Clark Kimberling, Jun 30 2015

Status

approved