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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). 60
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
T(n,k) is the number of k's in the partitions of n into equal parts. - 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
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.
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
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 *)
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
(PARI) {T(n, k) = if(n%k==0, n/k, 0)}; \\ G. C. Greubel, May 29 2019
(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
(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
CROSSREFS
Sequence in context: A309229 A143239 A158951 * A280499 A347351 A355342
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 March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)