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A127013 Triangle read by rows: row reversal of A126988.

%I A127013 #20 Sep 19 2024 04:15:43
%S A127013 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,
%T A127013 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,
%U A127013 0,0,0,0,0,2,0,3,4,6,12,1,0,0,0,0,0,0,0,0,0,0,0,13
%N A127013 Triangle read by rows: row reversal of A126988.
%C A127013 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).
%C A127013 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.
%C A127013 Row sums = sigma(n), A000203.
%D A127013 David Wells, "Prime Numbers, The Most Mysterious Figures in Math", John Wiley & Sons, 2005, Appendix.
%H A127013 Reinhard Zumkeller, <a href="/A127013/b127013.txt">Table of n, a(n) for n = 1..7875</a>
%e A127013 First few rows of the triangle are:
%e A127013    1;
%e A127013    1, 2;
%e A127013    1, 0, 3;
%e A127013    1, 0, 2, 4;
%e A127013    1, 0, 0, 0, 5;
%e A127013    1, 0, 0, 2, 3, 6;
%e A127013    1, 0, 0, 0, 0, 0, 7;
%e A127013    1, 0, 0, 0, 2, 0, 4, 8;
%e A127013    1, 0, 0, 0, 0, 0, 3, 0, 9;
%e A127013    1, 0, 0, 0, 0, 2, 0, 0, 5, 10;
%e A127013 Row 10 = (1, 0, 0, 0, 0, 2, 0, 0, 5, 10), reversal of 10th row of A126988.
%t A127013 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 *)
%o A127013 (Haskell)
%o A127013 a127013 n k = a127013_tabl !! (n-1) !! (k-1)
%o A127013 a127013_row n = a127013_tabl !! (n-1)
%o A127013 a127013_tabl = map reverse a126988_tabl
%o A127013 -- _Reinhard Zumkeller_, Jan 20 2014
%o A127013 (PARI) {T(n, k) = if(n%k==0, n/k, 0)};
%o A127013 for(n=1,12, for(k=1,n, print1(T(n,n-k+1), ", "))) \\ _G. C. Greubel_, Jun 03 2019
%o A127013 (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
%o A127013 (Sage)
%o A127013 def T(n, k):
%o A127013     if (n%k==0): return n/k
%o A127013     else: return 0
%o A127013 [[T(n, n-k+1) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jun 03 2019
%Y A127013 Cf. A126988, A000203.
%K A127013 nonn,tabl,easy
%O A127013 1,3
%A A127013 _Gary W. Adamson_, Jan 02 2007
%E A127013 T(10,10) fixed by _Reinhard Zumkeller_, Jan 20 2014
%E A127013 More terms added by _G. C. Greubel_, Jun 03 2019