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A290125 Square array read by antidiagonals T(n,k) = sigma(k + n) - sigma(k) - n, with n>=0 and k>=1.

%I A290125 #26 May 10 2021 02:36:27
%S A290125 0,0,1,0,0,1,0,2,2,3,0,-2,0,0,1,0,5,3,5,5,6,0,-5,0,-2,0,0,1,0,6,1,6,4,
%T A290125 6,6,7,0,-3,3,-2,3,1,3,3,4,0,4,1,7,2,7,5,7,7,8,0,-7,-3,-6,0,-5,0,-2,0,
%U A290125 0,1,0,15,8,12,9,15,10,15,13,15,15,16
%N A290125 Square array read by antidiagonals T(n,k) = sigma(k + n) - sigma(k) - n, with n>=0 and k>=1.
%C A290125 A015886(n) gives the position of the first zero in the n-th row of this array.
%H A290125 Michel Marcus, <a href="/A290125/b290125.txt">Table of n, a(n) for n = 0..5049</a>
%F A290125 T(0, k) = 0 for all k.
%e A290125 Array begins:
%e A290125   0, 0, 0,  0,  0,  0,  0, ...
%e A290125   1, 0, 2, -2,  5, -5,  6, ...
%e A290125   1, 2, 0,  3,  0,  1,  3, ...
%e A290125   3, 0, 5, -2,  6, -2,  7, ...
%e A290125   1, 5, 0,  4,  3,  2,  0, ...
%e A290125   6, 0, 6,  1,  7, -5, 15, ...
%e A290125   1, 6, 3,  5,  0, 10,  0, ...
%e A290125   7, 3, 7, -2, 15, -5,  9, ...
%e A290125   ...
%t A290125 Table[Function[n, If[k + n == 0, 0, DivisorSigma[1, k + n]] - If[k == 0, 0, DivisorSigma[1, k]] - n][m - k], {m, 12}, {k, m, 1, -1}] // Flatten (* _Michael De Vlieger_, Jul 20 2017 *)
%o A290125 (PARI) T(n,k) = sigma(k + n) - sigma(k) - n;
%o A290125 (PARI) a(n) = n++; my(s = ceil((-1+sqrt(1+8*n))/2));r=n-binomial(s,2)-1;k=s-r;T(r,k) \\ _David A. Corneth_, Jul 20 2017
%o A290125 (Python)
%o A290125 from sympy import divisor_sigma
%o A290125 l=[]
%o A290125 def T(n, k):
%o A290125     return 0 if n==0 or k==0 else divisor_sigma(k + n) - divisor_sigma(k) - n
%o A290125 for n in range(11): l+=[T(k, n - k + 1) for k in range(n + 1)]
%o A290125 print(l) # _Indranil Ghosh_, Jul 21 2017
%Y A290125 Cf. A000203 (sigma), A015886.
%K A290125 sign,tabl
%O A290125 0,8
%A A290125 _Michel Marcus_, Jul 20 2017