cp's OEIS Frontend

A299762 Irregular triangle T(n,k) read by rows in which row n lists the positive integers whose sum of divisors is n, or 0 if no such integer exists.

%I A299762 #61 Oct 13 2019 16:35:59
%S A299762 1,0,2,3,0,5,4,7,0,0,0,6,11,9,13,8,0,0,10,17,0,19,0,0,0,14,15,23,0,0,
%T A299762 0,12,0,29,16,25,21,31,0,0,0,22,0,37,18,27,0,20,26,41,0,43,0,0,0,33,
%U A299762 35,47,0,0,0,0,0,34,53,0,28,39,49,0,0,24,38,59,0,61,32,0,0,0,0,67,0,0,0,30,46,51,55,71,0,73
%N A299762 Irregular triangle T(n,k) read by rows in which row n lists the positive integers whose sum of divisors is n, or 0 if no such integer exists.
%C A299762 Essentially the same as the triangle described in the example section of A085790, but with 0's added in empty rows.
%C A299762 Are the records the same as A008578?
%F A299762 sigma(T(n,k)) = n, if T(n,k) >= 1.
%e A299762 First 24 rows of triangle T(n,k):
%e A299762 -----------------------
%e A299762 . n / k:  1   2   3 ...
%e A299762 -----------------------
%e A299762 | 1|      1;
%e A299762 | 2|      0;
%e A299762 | 3|      2;
%e A299762 | 4|      3;
%e A299762 | 5|      0;
%e A299762 | 6|      5;
%e A299762 | 7|      4;
%e A299762 | 8|      7;
%e A299762 | 9|      0;
%e A299762 |10|      0;
%e A299762 |11|      0;
%e A299762 |12|      6, 11;
%e A299762 |13|      9;
%e A299762 |14|     13;
%e A299762 |15|      8;
%e A299762 |16|      0;
%e A299762 |17|      0;
%e A299762 |18|     10, 17;
%e A299762 |19|      0;
%e A299762 |20|     19;
%e A299762 |21|      0;
%e A299762 |22|      0;
%e A299762 |23|      0;
%e A299762 |24|     14, 15, 23;
%e A299762 ...
%e A299762 For n = 23 there are no positive integers whose sum of divisors is 23, so T(23, 1) = 0, which is the only element in the 23rd row of the triangle.
%e A299762 For n = 24 there are three positive integers whose sum of divisors is 24; they are 14, 15 and 23, since sigma(14) = 1 + 2 + 7 + 14 = 24, sigma(15) = 1 + 3 + 5 + 15 = 24 and sigma(23) = 1 + 23 = 24, so the 24th row of the triangle is [14, 15, 23].
%t A299762 With[{nn = 74}, ReplacePart[ConstantArray[{0}, nn], PositionIndex@ Array[DivisorSigma[1, #] &, nn]]] // Flatten (* _Michael De Vlieger_, Mar 16 2018 *)
%Y A299762 Row sums give A258913.
%Y A299762 Column 1 gives A051444.
%Y A299762 Right border gives A057637.
%Y A299762 Positive terms give A085790.
%Y A299762 Row n has A054973(n) positive integers.
%Y A299762 Positive terms in the first column give A002192.
%Y A299762 Indices of the rows that contain a zero give A007369.
%Y A299762 Indices of the rows that contain positive terms give A002191.
%Y A299762 Cf. A000203, A007368, A007609, A008578.
%K A299762 nonn,tabf
%O A299762 1,3
%A A299762 _Omar E. Pol_, Mar 12 2018