A127013 Triangle read by rows: row reversal of A126988.

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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 0, 0, 0, 0, 0, 2, 0, 3, 4, 6, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13

Offset: 1

Author

Gary W. Adamson, Jan 02 2007

Keywords

• nonn
• tabl
• easy

Comments

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.

Examples

			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.
		

References

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

Links

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

Crossrefs

Cf. A126988, A000203.

Programs

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

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

Mathematica

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 *)

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

Sage

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

Extensions

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

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