1,3

Let j = reversed indices of row terms. Then for any row, j*T(n,k) = n, for nonzero T(n,k). For example, in row 10, we match the terms with their j indices: (1, 0, 0, 0, 0, 2, 0, 0, 5, 10), (dot product) (10, 9, 8, 7, 6, 5, 4, 3, 2, 1); getting (10, 0, 0, 0, 0, 10, 0, 0, 10, 10).

The factors of n are found in each row in order, as nonzero terms; e.g., 10 has the factors 1, 2, 5, 10, sum 18.

Row sums = sigma(n), A000203.

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..7875

First few rows of the triangle are:

1;

1, 2;

1, 0, 3;

1, 0, 2, 4;

1, 0, 0, 0, 5;

1, 0, 0, 2, 3, 6;

1, 0, 0, 0, 0, 0, 7;

1, 0, 0, 0, 2, 0, 4, 8;

1, 0, 0, 0, 0, 0, 3, 0, 9;

1, 0, 0, 0, 0, 2, 0, 0, 5, 10;

Row 10 = (1, 0, 0, 0, 0, 2, 0, 0, 5, 10), reversal of 10th row of A126988.

T[n_, m_]:= If[Mod[n, m]==0, n/m, 0]; Table[T[n, n-m+1], {n, 1, 12}, {m, 1, n}]//Flatten (* G. C. Greubel, Jun 03 2019 *)

(Haskell)

a127013 n k = a127013_tabl !! (n-1) !! (k-1)

a127013_row n = a127013_tabl !! (n-1)

a127013_tabl = map reverse a126988_tabl

-- Reinhard Zumkeller, Jan 20 2014

(PARI) {T(n, k) = if(n%k==0, n/k, 0)};

for(n=1, 12, for(k=1, n, print1(T(n, n-k+1), ", "))) \\ G. C. Greubel, Jun 03 2019

(Magma) [[(n mod (n-k+1)) eq 0 select n/(n-k+1) else 0: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 03 2019

(SageMath)

def T(n, k):

if (n%k==0): return n/k

else: return 0

[[T(n, n-k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 03 2019

Cf. A126988, A000203.

Sequence in context: A130055 A202452 A376789 * A117362 A247492 A113214

Adjacent sequences: A127010 A127011 A127012 * A127014 A127015 A127016

nonn,tabl,easy

Gary W. Adamson, Jan 02 2007

T(10,10) fixed by Reinhard Zumkeller, Jan 20 2014

More terms added by G. C. Greubel, Jun 03 2019

approved