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%I #29 Apr 02 2017 01:07:18
%S 1,1,3,1,5,1,3,7,1,9,1,3,4,13,1,13,1,3,7,15,1,5,19,1,3,10,17,1,21,1,3,
%T 4,5,11,28,1,25,1,3,16,21,1,5,7,41,1,3,7,15,31,1,33,1,3,4,13,6,59,1,
%U 37,1,3,7,3,31,21,1,5,13,53,1,3,28,29,1,45,1,3,4,5,11,4,36,39,1,9,61,1,3,34,33,1,5,19,65
%N Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th antidiagonal of the difference table of the divisors of n.
%C If n is prime then row n contains only two terms: 1 and 2*n-1.
%C Row 2^k gives the first k+1 positive terms of A000225, k >= 0.
%C Note that this sequence contains negative terms.
%C First differs from A274532 at a(41).
%e Triangle begins:
%e 1;
%e 1, 3;
%e 1, 5;
%e 1, 3, 7;
%e 1, 9;
%e 1, 3, 4, 13;
%e 1, 13;
%e 1, 3, 7, 15;
%e 1, 5, 19;
%e 1, 3, 10, 17;
%e 1, 21;
%e 1, 3, 4, 5, 11, 28;
%e 1, 25;
%e 1, 3, 16, 21;
%e 1, 5, 7, 41;
%e 1, 3, 7, 15, 31;
%e 1, 33;
%e 1, 3, 4, 13, 6, 59;
%e 1, 37;
%e 1, 3, 7, 3, 31, 21;
%e 1, 5, 13, 53;
%e 1, 3, 28, 29;
%e 1, 45;
%e 1, 3, 4, 5, 11, 4, 36, 39;
%e 1, 9, 61;
%e 1, 3, 34, 33;
%e 1, 5, 19, 65;
%e ...
%e For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, and the difference triangle of the divisors is
%e 1, 2, 3, 6, 9, 18;
%e 1, 1, 3, 3, 9;
%e 0, 2, 0, 6;
%e 2, -2, 6;
%e -4, 8;
%e 12;
%e The antidiagonal sums give [1, 3, 4, 13, 6, 59] which is also the 18th row of the irregular triangle.
%t Table[Map[Total, Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}], {1}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 27}] (* _Michael De Vlieger_, Jun 26 2016 *)
%o (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, i, sum(k=0, i-1, m[i-k, k+1]));}
%o tabf(nn) = for (n=1, nn, print(row(n)););
%o lista(nn) = for (n=1, nn, v = row(n); for (j=1, #v, print1(v[j], ", "));); \\ _Michel Marcus_, Jun 25 2016
%Y Row lengths give A000005. Column 1 is A000012. Right border gives A161700. Row sums give A273103.
%Y Cf. A000225, A161700, A187202, A272210, A273102, A273135, A273261, A273263, A274532.
%K sign,tabf
%O 1,3
%A _Omar E. Pol_, May 20 2016
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