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A119586 Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.

%I A119586 #23 May 24 2023 12:46:18
%S A119586 2,3,4,5,9,6,7,25,8,16,11,49,10,81,12,13,121,14,625,18,64,17,169,15,
%T A119586 2401,20,729,24,19,289,21,14641,28,15625,30,36,23,361,22,28561,32,
%U A119586 117649,40,100,48,29,529,26,83521,44,1771561,42,196,80,1024,31,841,27
%N A119586 Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.
%C A119586 From _Peter Munn_, May 17 2023: (Start)
%C A119586 As a square array A(n,m), n, m >= 1, read by ascending antidiagonals, A(n,m) is the n-th positive integer with m+1 divisors.
%C A119586 Thus both formats list the numbers with m+1 divisors in their m-th column. For the corresponding sequences giving numbers with a specific number of divisors see the index entries link.
%C A119586 (End)
%H A119586 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e A119586 Looking at the 4th row, 7 is the 4th positive integer with 2 divisors, 25 is the 3rd positive integer with 3 divisors, 8 is the 2nd positive integer with 4 divisors and 16 is the first positive integer with 5 divisors. So the 4th row is (7,25,8,16).
%e A119586 The triangle T(n,m) begins:
%e A119586   n\m:    1     2     3     4     5     6     7
%e A119586   ---------------------------------------------
%e A119586    1 :    2
%e A119586    2 :    3     4
%e A119586    3 :    5     9     6
%e A119586    4 :    7    25     8    16
%e A119586    5 :   11    49    10    81    12
%e A119586    6 :   13   121    14   625    18    64
%e A119586    7 :   17   169    15  2401    20   729    24
%e A119586   ...
%e A119586 Square array A(n,m) begins:
%e A119586   n\m:     1      2      3       4      5  ...
%e A119586   --------------------------------------------
%e A119586    1 :     2      4      6      16     12  ...
%e A119586    2 :     3      9      8      81     18  ...
%e A119586    3 :     5     25     10     625     20  ...
%e A119586    4 :     7     49     14    2401     28  ...
%e A119586    5 :    11    121     15   14641     32  ...
%e A119586   ...
%t A119586 t[n_, m_] := Block[{c = 0, k = 1}, While[c < n + 1 - m, k++; If[DivisorSigma[0, k] == m + 1, c++ ]]; k]; Table[ t[n, m], {n, 11}, {m, n}] // Flatten (* _Robert G. Wilson v_, Jun 07 2006 *)
%Y A119586 Columns: A000040, A001248, A007422, A030514, A030515, A030516, A030626, A030627, A030628, ... (see the index entries link for more).
%Y A119586 Cf. A073915.
%Y A119586 Diagonals (equivalently, rows of the square array) start: A005179\{1}, A161574.
%Y A119586 Cf. A091538.
%K A119586 nonn,tabl
%O A119586 1,1
%A A119586 _Leroy Quet_, May 31 2006
%E A119586 More terms from _Robert G. Wilson v_, Jun 07 2006