A132442 -id:A132442 - cp's OEIS frontend

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

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

Offset: 1

Clark Kimberling

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

			Rows:
  1;
  1, 2;
  1, 1, 3;
  1, 2, 1, 4;
  1, 1, 1, 1, 5;
  1, 2, 3, 2, 1, 6; ...

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

Cf. A003989.
Cf. A002262, A054531, A226314.
Cf. A018804 (row sums), A245717.
Cf. A132442 (sums of divisors).
Cf. A003418.

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

ColumnForm[Table[GCD[n, k], {k, 12}, {n, k}], Center] (* Alonso del Arte, Jan 14 2011 *)

{T(n, k) = gcd(n, k)} /* Michael Somos, Jul 18 2011 */

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

A134866 Table read by antidiagonals: T(n,k) = sigma(gcd(n,k)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 3, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 3, 1, 3, 6, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 7, 1, 12, 1, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 8, 3, 1, 3, 1, 3, 1

Offset: 1

Gary W. Adamson, Nov 14 2007

Previous name was: Triangle, antidiagonals of an array formed by A051731 * A127093 (transform).

Row sums give A094471.

			First few rows of the array:
  1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 1, 3, 1, 3, 1, ...
  1, 1, 4, 1, 1, 4, 1, ...
  1, 3, 1, 7, 1, 3, 1, ...
  1, 1, 1, 1, 6, 1, 1, ...
  ...
First antidiagonals:
  1;
  1, 1;
  1, 3, 1;
  1, 1, 1, 1;
  1, 3, 4, 3, 1;
  1, 1, 1, 1, 1, 1;
  1, 3, 1, 7, 1, 3, 1;
  1, 1, 4, 1, 1, 4, 1, 1;
  1, 3, 1, 3, 6, 3, 1, 3, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)

Cf. A051731, A127093, A094471, A132442.

Table[DivisorSigma[1, GCD[#, k]] &[n - k + 1], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Dec 19 2022 *)

T(n, k) = sigma(gcd(n, k)); \\ Michel Marcus, Dec 19 2022

T(n,k) = A000203(A050873(n,k)). - Michel Marcus, Dec 19 2022

New name and data corrected by Michel Marcus, Dec 19 2022

A265652 Triangle read by rows: T(n,k) is the sum of the union of the divisors of n and k.

Original entry on oeis.org

1, 3, 3, 4, 6, 4, 7, 7, 10, 7, 6, 8, 9, 12, 6, 12, 12, 12, 16, 17, 12, 8, 10, 11, 14, 13, 19, 8, 15, 15, 18, 15, 20, 24, 22, 15, 13, 15, 13, 19, 18, 21, 20, 27, 13, 18, 18, 21, 22, 18, 27, 25, 30, 30, 18, 12, 14, 15, 18, 17, 23, 19, 26, 24, 29, 12, 28, 28, 28, 28, 33, 28, 35, 36, 37, 43, 39, 28

Offset: 1

Franklin T. Adams-Watters, Dec 11 2015

Does every positive integer except 2 and 5 occur here? The stronger form of Goldbach's conjecture (every even integer > 6 is the sum of two distinct primes) suffices to show that every odd integer (except 5) is in the sequence, since T(p,q) = p + q + 1.

			Triangle begins:
   1
   3  3
   4  6  4
   7  7 10  7
   6  8  9 12  6
  12 12 12 16 17 12
  ...
The divisors of 3 are {1, 3}; the divisors of 4 are {1, 2, 4}. The union is {1, 2, 3, 4}, summing to 10; so T(4,3) = 10.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A000203 (first column and main diagonal).
T(2n,n) gives A062731.
Cf. A132442, A245093.

a265652 n k = a265652_tabl !! (n-1) !! (k-1)
a265652_row n = a265652_tabl !! (n-1)
a265652_tabl = zipWith (zipWith (-))
   (zipWith (map . (+)) a000203_list a245093_tabl) a132442_tabl
-- Reinhard Zumkeller, Dec 12 2015

seq(seq(numtheory:-sigma(n) + numtheory:-sigma(k) - numtheory:-sigma(igcd(n,k)), k=1..n), n=1..10); # Robert Israel, Dec 17 2015

Table[Total@ Union[Divisors@ n, Divisors@ k], {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Dec 18 2015 *)

T(n,k) = sigma(n) + sigma(k) - sigma(gcd(n,k))

T(n,k) = sigma(n) + sigma(k) - sigma(gcd(n,k)).

T(n,k) = A000203(n) + A245093(n,k) - A132442(n,k). - Reinhard Zumkeller, Dec 12 2015

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

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A134866 Table read by antidiagonals: T(n,k) = sigma(gcd(n,k)).

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A265652 Triangle read by rows: T(n,k) is the sum of the union of the divisors of n and k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula