|
%I #23 May 24 2023 12:46:18
%S 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 2401,20,729,24,19,289,21,14641,28,15625,30,36,23,361,22,28561,32,
%U 117649,40,100,48,29,529,26,83521,44,1771561,42,196,80,1024,31,841,27
%N Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.
%C From _Peter Munn_, May 17 2023: (Start)
%C 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 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 (End)
%H <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e 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 The triangle T(n,m) begins:
%e n\m: 1 2 3 4 5 6 7
%e ---------------------------------------------
%e 1 : 2
%e 2 : 3 4
%e 3 : 5 9 6
%e 4 : 7 25 8 16
%e 5 : 11 49 10 81 12
%e 6 : 13 121 14 625 18 64
%e 7 : 17 169 15 2401 20 729 24
%e ...
%e Square array A(n,m) begins:
%e n\m: 1 2 3 4 5 ...
%e --------------------------------------------
%e 1 : 2 4 6 16 12 ...
%e 2 : 3 9 8 81 18 ...
%e 3 : 5 25 10 625 20 ...
%e 4 : 7 49 14 2401 28 ...
%e 5 : 11 121 15 14641 32 ...
%e ...
%t 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 Columns: A000040, A001248, A007422, A030514, A030515, A030516, A030626, A030627, A030628, ... (see the index entries link for more).
%Y Cf. A073915.
%Y Diagonals (equivalently, rows of the square array) start: A005179\{1}, A161574.
%Y Cf. A091538.
%K nonn,tabl
%O 1,1
%A _Leroy Quet_, May 31 2006
%E More terms from _Robert G. Wilson v_, Jun 07 2006
|
