A127446 -id:A127446 - cp's OEIS frontend

A143308 Triangle read by rows, A127446 * A000012, 1<=k<=n.

1, 4, 2, 6, 3, 3, 12, 8, 4, 4, 10, 5, 5, 5, 5, 24, 18, 12, 6, 6, 6, 14, 7, 7, 7, 7, 7, 7, 32, 24, 16, 16, 8, 8, 8, 8, 27, 18, 18, 9, 9, 9, 9, 9, 9, 40, 30, 20, 20, 20, 10, 10, 10, 10, 10, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 72, 60, 48, 36, 24, 24, 12, 12, 12, 12, 12, 12

Offset: 1

Gary W. Adamson, Aug 06 2008

Given triangle A127446, partial sums of terms starting from the right.
Row sums give A064987.
Left column is A038040.

			First few rows of the triangle =
   1;
   4,  2;
   6,  3,  3;
  12,  8,  4,  4;
  10,  5,  5,  5,  5;
  24, 18, 12,  6,  6,  6;
  14,  7,  7,  7,  7,  7,  7;
Row 4 = (12, 8, 4, 4) since row 4 of triangle A127446 = (4, 4, 0, 4).

Cf. A038040, A064987, A127446.

a(56) corrected by Georg Fischer, Jul 04 2023

A127447 Triangle defined by the matrix product A127446 * A054521, read by rows 1<=k<=n.

Original entry on oeis.org

1, 4, 0, 6, 3, 0, 12, 0, 4, 0, 10, 5, 5, 5, 0, 24, 6, 0, 0, 6, 0, 14, 7, 7, 7, 7, 7, 0, 32, 0, 16, 0, 8, 0, 8, 0, 27, 18, 0, 9, 9, 0, 9, 9, 0, 40, 10, 20, 10, 0, 0, 10, 0, 10, 0, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 0, 72, 12, 12, 0, 24, 0

Offset: 1

Gary W. Adamson, Jan 14 2007

			First few rows of the triangle are:
1;
4, 0;
6, 3, 0;
12, 0, 4, 0;
10, 5, 5, 5, 0;
24, 6, 0, 0, 6, 0;
14, 7, 7, 7, 7, 7, 0;
32, 0, 16, 0, 8 0, 8, 0;
...

Cf. A038040 (column k=1), A000290 (row sums), A127446, A054521.

A054521 := proc(n, k) if igcd(n,k) = 1 then 1; else 0; fi; end:
A127447 := proc(n,k)
        add( A127446(n,j)*A054521(j,k),j=k..n) ;
end proc:
seq(seq(A127447(n,m),m=1..n),n=1..12) ; # R. J. Mathar, Nov 08 2011

T(n,k) = Sum_{j=k..n} A127446(n,j) * A054521(j,k).

A143310 Triangle read by rows, A000012 * A127446, 1 <= k <= n.

Original entry on oeis.org

1, 3, 2, 6, 2, 3, 10, 6, 3, 4, 15, 6, 3, 4, 5, 21, 12, 9, 4, 5, 6, 28, 12, 9, 4, 5, 6, 7, 36, 20, 9, 12, 5, 6, 7, 8, 45, 20, 18, 12, 5, 6, 7, 8, 9, 55, 30, 18, 12, 15, 6, 7, 8, 9, 10, 66, 30, 18, 12, 15, 6, 7, 8, 9, 10, 11, 78, 42, 30, 24, 15, 18, 7, 8, 9, 10, 11, 12

Offset: 1

Gary W. Adamson, Aug 06 2008

Row sums = A143127: (1, 5, 11, 23, 33, 57, 71, ...).

			First few rows of the triangle:
   1;
   3,  2;
   6,  2,  3;
  10,  6,  3,  4;
  15,  6,  3,  4,  5;
  21, 12,  9,  4,  5,  6;
  28, 12,  9,  4,  5,  6,  7;
  36, 20,  9, 12,  5,  6,  7,  8;
  ...

Cf. A127446, A143127.

Triangle read by rows, A000012 * A127446, 1 <= k <= n.

Conjecture: T(n,k) = Sum_{i=1..n} [i*k <= n]*i*k. - Mats Granvik, Feb 26 2021

Duplicate a(11)-a(15) terms removed by Mats Granvik, Feb 26 2021

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).

Original entry on oeis.org

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

Gary W. Adamson, Dec 31 2006

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

			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

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

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

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

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

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

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

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 *)

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

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

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

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

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

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A143308 Triangle read by rows, A127446 * A000012, 1<=k<=n.

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A127447 Triangle defined by the matrix product A127446 * A054521, read by rows 1<=k<=n.

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A143310 Triangle read by rows, A000012 * A127446, 1 <= k <= n.

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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).

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