

A259573


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


3



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
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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, AddisonWesley, 1989, pages 9094.


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



