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

1,3

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.

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

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Cf. A126988, A000203.

Sequence in context: A130055 A202452 A376789 * A117362 A247492 A113214

Adjacent sequences: A127010 A127011 A127012 * A127014 A127015 A127016

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Jan 02 2007

Extensions

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

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

Status

approved