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%I #36 Jan 23 2023 09:04:52
%S 1,2,4,3,6,9,5,8,12,16,7,10,15,20,25,11,14,18,24,30,36,13,22,21,28,35,
%T 42,49,17,26,27,32,40,48,56,64,19,34,33,44,45,54,63,72,81,23,38,39,52,
%U 50,60,70,80,90,100,29,46,51,68,55,66,77,88,99,110,121,31,58,57,76,65,78,84,96,108,120,132,144
%N Square array read by antidiagonals where column k lists the numbers j whose largest divisor <= sqrt(j) is k.
%C This sequence is a permutation of the natural numbers A000027. Note that the first column is formed by 1 together with the prime numbers.
%C Column k contains exactly those numbers j=k*m where m is either a prime >= j or one of the numbers in row k of A163925. - _Franklin T. Adams-Watters_, Aug 12 2009
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv07.jpg">Illustration of initial terms of column 1: A008578</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv08.jpg">Illustration of initial terms of column 2: A161344</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv06.jpg">Illustration of initial terms of columns 1-4: A008578, A161344, A161345, A161424</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F Column k lists the numbers j such that A033676(j)=k.
%e Array begins:
%e 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ...
%e 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, ...
%e 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, ...
%e 5, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, ...
%e 7, 14, 21, 32, 45, 60, 77, 96, 117, 140, 165, 192, ...
%e 11, 22, 27, 44, 50, 66, 84, 104, 126, 150, 176, 204, ...
%e 13, 26, 33, 52, 55, 78, 91, 112, 135, 160, 187, 216, ...
%e 17, 34, 39, 68, 65, 102, 98, 128, 153, 170, 198, 228, ...
%e 19, 38, 51, 76, 75, 114, 105, 136, 162, 190, 209, 264, ...
%e 23, 46, 57, 92, 85, 138, 119, 152, 171, 200, 220, 276, ...
%e 29, 58, 69, 116, 95, 174, 133, 184, 189, 230, 231, 348, ...
%e 31, 62, 87, 124, 115, 186, 147, 232, 207, 250, 242, 372, ...
%e ...
%p A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: # _R. J. Mathar_, Aug 09 2009
%t nmax = 12;
%t pm = Prime[nmax];
%t sDiv[n_] := Select[Divisors[n], #^2 <= n&][[-1]];
%t Clear[col]; col[k_] := col[k] = Select[Range[k pm], sDiv[#] == k&];
%t T[n_, k_ /; 1 <= k <= Length[col[k]]] := col[k][[n]];
%t Table[T[n-k+1, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 15 2019 *)
%Y Rows 1-12: A000290, A002378, A005563, A164004, A100451, A164006, A164007, A164008, A164009, A164010, A164011, A164012.
%Y Columns 1-12: A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
%Y Another version: A163990.
%Y Cf. A000027, A000040, A033676, A147861, A163100, A164000.
%K nonn,tabl
%O 1,2
%A _Omar E. Pol_, Aug 07 2009
%E Edited by _R. J. Mathar_, Aug 01 2010
%E Example edited by _Jean-François Alcover_, Dec 15 2019