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A050873 Triangular array T read by rows: T(n,k) = gcd(n,k).

%I A050873 #58 Feb 16 2025 08:32:40
%S A050873 1,1,2,1,1,3,1,2,1,4,1,1,1,1,5,1,2,3,2,1,6,1,1,1,1,1,1,7,1,2,1,4,1,2,
%T A050873 1,8,1,1,3,1,1,3,1,1,9,1,2,1,2,5,2,1,2,1,10,1,1,1,1,1,1,1,1,1,1,11,1,
%U A050873 2,3,4,1,6,1,4,3,2,1,12,1,1,1,1,1,1,1,1,1,1
%N A050873 Triangular array T read by rows: T(n,k) = gcd(n,k).
%C A050873 The function T(n,k) = T(k,n) is defined for all integer k,n but only the values for 1 <= k <= n as a triangular array are listed here.
%C A050873 For each divisor d of n, the number of d's in row n is phi(n/d). Furthermore, if {a_1, a_2, ..., a_phi(n/d)} is the set of positive integers <= n/d that are relatively prime to n/d then T(n,a_i * d) = d. - _Geoffrey Critzer_, Feb 22 2015
%C A050873 Starting with any row n and working downwards, consider the infinite rectangular array with k = 1..n. A repeating pattern occurs every A003418(n) rows. For example, n=3: A003418(3) = 6. The 6-row pattern starting with row 3 is {1,1,3}, {1,2,1}, {1,1,1}, {1,2,3}, {1,1,1}, {1,2,1}, and this pattern repeats every 6 rows, i.e., starting with rows {9,15,21,27,...}. - _Bob Selcoe_ and _Jamie Morken_, Aug 02 2017
%H A050873 T. D. Noe, <a href="/A050873/b050873.txt">Rows n=1..100, flattened</a>
%H A050873 Marcelo Polezzi, <a href="http://www.jstor.org/stable/2974739">A Geometrical Method for Finding an Explicit Formula for the Greatest Common Divisor</a>, The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), pp. 445-446.
%H A050873 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GreatestCommonDivisor.html">Greatest Common Divisor</a>
%H A050873 Wikipedia, <a href="http://en.wikipedia.org/wiki/Greatest_common_divisor">Greatest Common Divisor</a>
%F A050873 a(n) = gcd(A002260(n), A002024(n)); A054521(n) = A000007(a(n)). - _Reinhard Zumkeller_, Dec 02 2009
%F A050873 T(n,k) = A075362(n,k)/A051173(n,k), 1 <= k <= n. - _Reinhard Zumkeller_, Apr 25 2011
%F A050873 T(n, k) = T(k, n) = T(-n, k) = T(n, -k) = T(n, n+k) = T(n+k, k). - _Michael Somos_, Jul 18 2011
%F A050873 T(n,k) = A051173(n,k) / A051537(n,k). - _Reinhard Zumkeller_, Jul 07 2013
%e A050873 Rows:
%e A050873   1;
%e A050873   1, 2;
%e A050873   1, 1, 3;
%e A050873   1, 2, 1, 4;
%e A050873   1, 1, 1, 1, 5;
%e A050873   1, 2, 3, 2, 1, 6; ...
%t A050873 ColumnForm[Table[GCD[n, k], {k, 12}, {n, k}], Center] (* _Alonso del Arte_, Jan 14 2011 *)
%o A050873 (PARI) {T(n, k) = gcd(n, k)} /* _Michael Somos_, Jul 18 2011 */
%o A050873 (Haskell)
%o A050873 a050873 = gcd
%o A050873 a050873_row n = a050873_tabl !! (n-1)
%o A050873 a050873_tabl = zipWith (map . gcd ) [1..] a002260_tabl
%o A050873 -- _Reinhard Zumkeller_, Dec 12 2015, Aug 13 2013, Jun 10 2013
%Y A050873 Cf. A003989.
%Y A050873 Cf. A002262, A054531, A226314.
%Y A050873 Cf. A018804 (row sums), A245717.
%Y A050873 Cf. A132442 (sums of divisors).
%Y A050873 Cf. A003418.
%K A050873 nonn,tabl,look
%O A050873 1,3
%A A050873 _Clark Kimberling_