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A273263 Irregular triangle read by rows: T(n,k) is the sum of the elements of the k-th column of the difference table of the divisors of n.

%I A273263 #30 Aug 31 2021 02:43:03
%S A273263 1,2,2,3,3,3,4,4,5,5,4,5,6,6,7,7,4,6,8,8,7,9,9,4,7,10,10,11,11,4,6,8,
%T A273263 10,12,12,13,13,4,9,14,14,11,13,15,15,5,8,12,16,16,17,17,12,11,12,15,
%U A273263 18,18,19,19,-3,4,10,15,20,20,13,17,21,21,4,13,22,22,23,23,-4,3,8,12,16,20,24,24,21,25,25
%N A273263 Irregular triangle read by rows: T(n,k) is the sum of the elements of the k-th column of the difference table of the divisors of n.
%C A273263 If n is prime then row n is [n, n].
%C A273263 It appears that the last two terms of the n-th row are [n, n], n > 1.
%C A273263 First differs from A274533 at a(38).
%e A273263 Triangle begins:
%e A273263    1;
%e A273263    2,  2;
%e A273263    3,  3;
%e A273263    3,  4,  4;
%e A273263    5,  5;
%e A273263    4,  5,  6,  6;
%e A273263    7,  7;
%e A273263    4,  6,  8,  8;
%e A273263    7,  9,  9;
%e A273263    4,  7, 10, 10;
%e A273263   11, 11;
%e A273263    4,  6,  8, 10, 12, 12;
%e A273263   13, 13;
%e A273263    4,  9, 14, 14;
%e A273263   11, 13, 15, 15;
%e A273263    5,  8, 12, 16, 16;
%e A273263   17, 17;
%e A273263   12, 11, 12, 15, 18, 18;
%e A273263   19, 19;
%e A273263   -3,  4, 10, 15, 20, 20;
%e A273263   13, 17, 21, 21;
%e A273263    4, 13, 22, 22;
%e A273263   23, 23;
%e A273263   -4,  3,  8, 12, 16, 20, 24, 24;
%e A273263   21, 25, 25;
%e A273263    4, 15, 26, 26;
%e A273263   ...
%e A273263 For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, and the difference triangle of the divisors is
%e A273263    1,  2,  3,  6,  9, 18;
%e A273263    1,  1,  3,  3,  9;
%e A273263    0,  2,  0,  6;
%e A273263    2, -2,  6;
%e A273263   -4,  8;
%e A273263   12;
%e A273263 The column sums give [12, 11, 12, 15, 18, 18] which is also the 18th row of the irregular triangle.
%t A273263 Table[Total /@ Transpose@ Map[Function[w, PadRight[w, Length@ #]], NestWhileList[Differences, #, Length@ # > 1 &]] &@ Divisors@ n, {n, 25}] // Flatten (* _Michael De Vlieger_, Jun 26 2016 *)
%o A273263 (PARI) row(n) = {my(d = divisors(n)); my(nd = #d); my(m = matrix(#d, #d)); for (j=1, nd, m[1,j] = d[j];); for (i=2, nd, for (j=1, nd - i +1, m[i,j] = m[i-1,j+1] - m[i-1,j];);); vector(nd, j, sum(i=1, nd, m[i, j]));}
%o A273263 tabf(nn) = for (n=1, nn, print(row(n)););
%o A273263 lista(nn) = for (n=1, nn, v = row(n); for (j=1, #v, print1(v[j], ", "));); \\ _Michel Marcus_, Jun 25 2016
%Y A273263 Row lengths give A000005. Right border gives A000027. Column 1 is A161857. Row sums give A273103.
%Y A273263 Cf. A187202, A273102, A273135, A272210, A273136, A273261, A273262, A274533.
%K A273263 sign,tabf
%O A273263 1,2
%A A273263 _Omar E. Pol_, May 22 2016