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A259576
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Number of distinct differences in row n of the reciprocity array of 1.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(PARI)
A259575sq(m, n) = sum(k=0, m-1, (1+(n*k))\m);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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