A126988 Triangle read by rows: T(n,k) = n/k if k is a divisor of n; T(n,k) = 0 if k is not a divisor of n (1 <= k <= n).

1, 2, 1, 3, 0, 1, 4, 2, 0, 1, 5, 0, 0, 0, 1, 6, 3, 2, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 8, 4, 0, 2, 0, 0, 0, 1, 9, 0, 3, 0, 0, 0, 0, 0, 1, 10, 5, 0, 0, 2, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Gary W. Adamson, Dec 31 2006

Keywords

• nonn
• easy
• tabl

Comments

Row sums = A000203, sigma(n).

k-th column (k=0,1,2,...) is (1,2,3,...) interspersed with n consecutive zeros starting after the "1".

The nonzero entries of row n are the divisors of n in decreasing order. - Emeric Deutsch, Jan 17 2007

Alternating row sums give A000593. - Omar E. Pol, Feb 11 2018

T(n,k) is the number of k's in the partitions of n into equal parts. - Omar E. Pol, Nov 25 2019

Examples

			First few rows of the triangle are:
   1;
   2, 1;
   3, 0, 1;
   4, 2, 0, 1;
   5, 0, 0, 0, 1;
   6, 3, 2, 0, 0, 1;
   7, 0, 0, 0, 0, 0, 1;
   8, 4, 0, 2, 0, 0, 0, 1;
   9, 0, 3, 0, 0, 0, 0, 0, 1;
  10, 5, 0, 0, 2, 0, 0, 0, 0, 1;
  ...
sigma(12) = A000203(n) = 28.
sigma(12) = 28, from 12th row = (12 + 6 + 4 + 3 + 2 + 1), deleting the zeros, from left to right.
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the number of k's are [6, 3, 2, 0, 0, 1] respectively, equaling the 6th row of triangle. - _Omar E. Pol_, Nov 25 2019
		

References

• David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John Wiley & Sons, Inc, 2005, Appendix B.

Links

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

Crossrefs

Cf. A000203, A008284, A127446, A244051, A328361.

Programs

Haskell

a126988 n k = a126988_tabl !! (n-1) !! (k-1)
a126988_row n = a126988_tabl !! (n-1)
a126988_tabl = zipWith (zipWith (*)) a010766_tabl a051731_tabl
-- Reinhard Zumkeller, Jan 20 2014

Magma

[[(n mod k) eq 0 select n/k else 0: k in [1..n]]: n in [1..12]]; // G. C. Greubel, May 29 2019

Maple

A126988:=proc(n,k) if type(n/k, integer)=true then n/k else 0 fi end: for n from 1 to 12 do seq(A126988(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Jan 17 2007

Mathematica

Table[If[Mod[n, m]==0, n/m, 0], {n,1,12}, {m,1,n}]//Flatten (* Roger L. Bagula, Sep 06 2008, simplified by Franklin T. Adams-Watters, Aug 24 2011 *)

PARI

{T(n,k) = if(n%k==0, n/k, 0)}; \\ G. C. Greubel, May 29 2019

Sage

def T(n, k):
    if (n%k==0): return n/k
    else: return 0
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 29 2019

Formula

From Emeric Deutsch, Jan 17 2007: (Start)

G.f. of column k: z^k/(1-z^k)^2 (k=1,2,...).

G.f.: G(t,z) = Sum_{k>=1} t^k*z^k/(1-z^k)^2. (End)

G.f.: F(x,z) = log(1/(Product_{n >= 1} (1 - x*z^n))) = Sum_{n >= 1} (x*z)^n/(n*(1 - z^n)) = x*z + (2*x + x^2)*z^2/2 + (3*x + x^3)*z^3/3 + .... Note, exp(F(x,z)) is a g.f. for A008284 (with an additional term T(0,0) equal to 1). - Peter Bala, Jan 13 2015

T(n,k) = A010766(n,k)*A051731(n,k), k=1..n. - Reinhard Zumkeller, Jan 20 2014

Extensions

Edited by N. J. A. Sloane, Jan 24 2007

Comment from Emeric Deutsch made name by Franklin T. Adams-Watters, Aug 24 2011