A273263 - OEIS

%I #30 Aug 31 2021 02:43:03

%S 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 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 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 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 If n is prime then row n is [n, n].

%C It appears that the last two terms of the n-th row are [n, n], n > 1.

%C First differs from A274533 at a(38).

%e Triangle begins:

%e    1;

%e    2,  2;

%e    3,  3;

%e    3,  4,  4;

%e    5,  5;

%e    4,  5,  6,  6;

%e    7,  7;

%e    4,  6,  8,  8;

%e    7,  9,  9;

%e    4,  7, 10, 10;

%e   11, 11;

%e    4,  6,  8, 10, 12, 12;

%e   13, 13;

%e    4,  9, 14, 14;

%e   11, 13, 15, 15;

%e    5,  8, 12, 16, 16;

%e   17, 17;

%e   12, 11, 12, 15, 18, 18;

%e   19, 19;

%e   -3,  4, 10, 15, 20, 20;

%e   13, 17, 21, 21;

%e    4, 13, 22, 22;

%e   23, 23;

%e   -4,  3,  8, 12, 16, 20, 24, 24;

%e   21, 25, 25;

%e    4, 15, 26, 26;

%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 column sums give [12, 11, 12, 15, 18, 18] which is also the 18th row of the irregular triangle.

%t 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 (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 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. Right border gives A000027. Column 1 is A161857. Row sums give A273103.

%Y Cf. A187202, A273102, A273135, A272210, A273136, A273261, A273262, A274533.

%K sign,tabf

%O 1,2

%A _Omar E. Pol_, May 22 2016