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 A050873 Triangular array T read by rows: T(n,k) = gcd(n,k). 43
 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, 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, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. 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 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 LINKS T. D. Noe, Rows n=1..100, flattened 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. 445-446. Eric Weisstein's World of Mathematics, Greatest Common Divisor Wikipedia, Greatest Common Divisor FORMULA a(n) = gcd(A002260(n), A002024(n)); A054521(n) = A000007(a(n)). - Reinhard Zumkeller, Dec 02 2009 T(n,k) = A075362(n,k)/A051173(n,k), 1 <= k <= n. - Reinhard Zumkeller, Apr 25 2011 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 T(n,k) = A051173(n,k) / A051537(n,k). - Reinhard Zumkeller, Jul 07 2013 EXAMPLE Rows: 1; 1, 2; 1, 1, 3; 1, 2, 1, 4; 1, 1, 1, 1, 5; 1, 2, 3, 2, 1, 6; ... MATHEMATICA ColumnForm[Table[GCD[n, k], {k, 12}, {n, k}], Center] (* Alonso del Arte, Jan 14 2011 *) PROG (PARI) {T(n, k) = gcd(n, k)} /* Michael Somos, Jul 18 2011 */ (Haskell) a050873 = gcd a050873_row n = a050873_tabl !! (n-1) a050873_tabl = zipWith (map . gcd ) [1..] a002260_tabl -- Reinhard Zumkeller, Dec 12 2015, Aug 13 2013, Jun 10 2013 CROSSREFS Cf. A003989. Cf. A002262, A054531, A226314. Cf. A018804 (row sums), A245717. Cf. A132442 (sums of divisors). Cf. A003418. Sequence in context: A277760 A127704 A307662 * A324668 A128221 A327856 Adjacent sequences: A050870 A050871 A050872 * A050874 A050875 A050876 KEYWORD nonn,tabl,look AUTHOR STATUS approved

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Last modified November 29 13:48 EST 2022. Contains 358430 sequences. (Running on oeis4.)