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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). 55
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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

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

EXAMPLE

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.

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

t[n_, m_] = If[Mod[n, m] == 0, n/m, 0]; Table[Table[t[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[%] (* Roger L. Bagula, Sep 06 2008, simplified by Franklin T. Adams-Watters, Aug 24 2011 *)

PROG

(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

CROSSREFS

Cf. A000203, A008284.

Sequence in context: A158906 A143239 A158951 * A280499 A130026 A113287

Adjacent sequences:  A126985 A126986 A126987 * A126989 A126990 A126991

KEYWORD

nonn,easy,tabl

AUTHOR

Gary W. Adamson, Dec 31 2006

EXTENSIONS

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

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

STATUS

approved

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Last modified January 18 11:33 EST 2019. Contains 319271 sequences. (Running on oeis4.)