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A259576 Number of distinct differences in row n of the reciprocity array of 1. 3
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, 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, 11, 10, 9, 14, 3, 10, 5, 16, 3, 18, 3, 6, 9, 10, 7, 14, 3 (list; graph; refs; listen; history; text; internal format)
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).  See A259572 for a guide to related sequences.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..79.

EXAMPLE

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.

MATHEMATICA

x = 1;  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}]  (* A259576 *)

CROSSREFS

Cf. A249572, A249573, A259575.

Sequence in context: A169809 A214962 A076258 * A030330 A286579 A293689

Adjacent sequences:  A259573 A259574 A259575 * A259577 A259578 A259579

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 01 2015

STATUS

approved

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Last modified October 20 05:42 EDT 2017. Contains 293601 sequences.