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A056538 Irregular triangle read by rows: row n lists the divisors of n in decreasing order.

%I A056538 #30 Jul 02 2025 23:16:43
%S A056538 1,2,1,3,1,4,2,1,5,1,6,3,2,1,7,1,8,4,2,1,9,3,1,10,5,2,1,11,1,12,6,4,3,
%T A056538 2,1,13,1,14,7,2,1,15,5,3,1,16,8,4,2,1,17,1,18,9,6,3,2,1,19,1,20,10,5,
%U A056538 4,2,1,21,7,3,1,22,11,2,1,23,1,24,12,8,6,4,3,2,1,25,5,1,26,13,2,1,27,9
%N A056538 Irregular triangle read by rows: row n lists the divisors of n in decreasing order.
%C A056538 Old name was "Replace n by its divisors in reverse order."
%C A056538 This gives the second elements of the ordered pairs (a,b), a >= 1, b >= 1, ordered by their product ab.
%C A056538 T(n,k) = n / A027750(n,k) = A027750(n,n-k+1), 1 <= k <= A000005(n). - _Reinhard Zumkeller_, Sep 28 2014
%C A056538 The 2nd column of the triangle is the largest proper divisor (A032742). - _Charles Kusniec_, Jan 30 2021
%H A056538 Reinhard Zumkeller, <a href="/A056538/b056538.txt">Rows n = 1..1000 of triangle, flattened</a>
%H A056538 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv05.jpg">Illustration of the divisors of n</a> - _Omar E. Pol_, Nov 22 2009
%F A056538 a(n) = A064894(A064896(n)).
%e A056538 Triangle begins:
%e A056538 1;
%e A056538 2, 1;
%e A056538 3, 1;
%e A056538 4, 2, 1;
%e A056538 5, 1;
%e A056538 6, 3, 2, 1;
%e A056538 7, 1;
%e A056538 8, 4, 2, 1;
%e A056538 9, 3, 1;
%e A056538 10, 5, 2, 1;
%e A056538 11, 1;
%e A056538 12, 6, 4, 3, 2, 1;
%e A056538 13, 1;
%e A056538 14, 7, 2, 1;
%e A056538 15, 5, 3, 1;
%e A056538 16, 8, 4, 2, 1;
%e A056538 17, 1;
%e A056538 18, 9, 6, 3, 2, 1;
%e A056538 19, 1;
%e A056538 20, 10, 5, 4, 2, 1;
%p A056538 map(op,[seq(reverse(sort(divisors(j))),j=1..30)]);
%p A056538 cdr := proc(l) if 0 = nops(l) then ([]) else (l[2..nops(l)]): fi: end:
%p A056538 reverse := proc(l) if 0 = nops(l) then ([]) else [op(reverse(cdr(l))), l[1]]; fi: end:
%t A056538 Table[Reverse@ Divisors@ n, {n, 27}] // Flatten (* _Michael De Vlieger_, Jul 27 2016 *)
%o A056538 (Magma) [Reverse(Divisors(n)) : n in [1..30]];
%o A056538 (Haskell)
%o A056538 a056538 n k = a056538_tabf !! (n-1) !! (k-1)
%o A056538 a056538_row n = a056538_tabf !! (n-1)
%o A056538 a056538_tabf = map reverse a027750_tabf
%o A056538 -- _Reinhard Zumkeller_, Sep 28 2014
%o A056538 (PARI) row(n)=Vecrev(divisors(n)) \\ _Charles R Greathouse IV_, Sep 02 2015
%Y A056538 Cf. A027750 for the first elements, A056534, A168017, A000005 (row lengths), A000203 (row sums), A032742 (largest proper divisor).
%K A056538 nonn,tabf
%O A056538 1,2
%A A056538 _Antti Karttunen_, Jun 20 2000
%E A056538 Definition revised by _N. J. A. Sloane_, Jul 27 2016