

A003989


Table of gcd(x,y) read by antidiagonals, where (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...


39



1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 1, 2, 1
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OFFSET

1,5


COMMENTS

For m < n, the maximal number of nonattacking queens that can be placed on the n by m rectangular toroidal chessboard is gcd(m,n), except in the case m=3, n=6.
The determinant of the matrix of the first n rows and columns is A001088(n). [Smith, Mansion]  Michael Somos, Jun 25 2012
Imagine a torus having regular polygonal crosssection of m sides. Now, break the torus and twist the free ends, preserving rotational symmetry, then reattach the ends. Let n be the number of faces passed in twisting the torus before reattaching it. For example, if n = m, then the torus has exactly one full twist. Do this for arbitrary m and n (m > 1, n > 0). Now, count the number of independent, closed paths on the surface of the resulting torus, where a path is "closed" if and only if it returns to its starting point after a finite number of times around the surface of the torus. Conjecture: this number is always gcd(m,n). NOTE: This figure constitutes a group with m and n the binary arguments and gcd(m,n) the resulting value. Twisting in the reverse direction is the inverse operation, and breaking & reattaching in place is the identity operation.  Jason RichardsonWhite, May 06 2013
Regarded as a triangle, table of gcd(n,k) for 0 < k < n.  Franklin T. AdamsWatters, Oct 09 2014


REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, AddisonWesley, 2nd ed., 1994, ch. 4.
D. E. Knuth, The Art of Computer Programming, AddisonWesley, section 4.5.2.


LINKS

T. D. Noe, First 100 antidiagonals of array, flattened
Grant Cairns, Queens on Nonsquare Tori, El. J. Combinatorics, N6, 2001.
P. Mansion, On an Arithmetical Theorem of Professor Smith's, Messenger of Mathematics, (1878), pp. 8182.
Kival Ngaokrajang, Pattern of GCD(x,y) > 1 for x and y = 1..60. Nonisolated values larger than 1 (polyomino shapes) are colored.
Marcelo Polezzi, A Geometrical Method for Finding an Explicit Formula for the Greatest Common Divisor, The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), pp. 445446.
H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (18751876), pp. 208212.
Index entries for sequences related to lcm's


FORMULA

Multiplicative in both parameters with a(p^e, m) = GCD(p^e, m).  David W. Wilson, Jun 12 2005


EXAMPLE

Table begins:
1 1 1 1 1 1 ...
1 2 1 2 1 2 ...
1 1 3 1 1 3 ...
1 2 1 4 1 2 ...
1 1 1 1 5 1 ...


MATHEMATICA

Table[ GCD[x  y + 1, y], {x, 1, 15}, {y, 1, x}] // Flatten (* JeanFrançois Alcover, Dec 12 2012 *)


PROG

(PARI) {A(n, m) = gcd(n, m)}; /* Michael Somos, Jun 25 2012 */


CROSSREFS

Rows, columns and diagonals: A089128, A109007, A109008, A109009, A109010, A109011, A109012, A109013, A109014, A109015.
A109004 is (0, 0) based.
Cf. A003990, A003991, A050873, A054431, A001088.
Cf. also A091255 for GF(2)[X] polynomial analog.
A(x, y) = A075174(A004198(A075173(x), A075173(y))) = A075176(A004198(A075175(x), A075175(y))).
Antidiagonal sums are in A006579.
Sequence in context: A140194 A159923 A287957 * A091255 A175466 A214403
Adjacent sequences: A003986 A003987 A003988 * A003990 A003991 A003992


KEYWORD

tabl,nonn,easy,nice,mult


AUTHOR

Marc LeBrun


STATUS

approved



