A328524 T(n,k) is the k-th smallest least integer of prime signatures for partitions of n into distinct parts; triangle T(n,k), n>=0, 1<=k<=A000009(n), read by rows.

1, 2, 4, 8, 12, 16, 24, 32, 48, 72, 64, 96, 144, 360, 128, 192, 288, 432, 720, 256, 384, 576, 864, 1440, 2160, 512, 768, 1152, 1728, 2592, 2880, 4320, 10800, 1024, 1536, 2304, 3456, 5184, 5760, 8640, 12960, 21600, 75600, 2048, 3072, 4608, 6912, 10368, 11520

Offset: 0

Author

Alois P. Heinz, Feb 18 2020

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
     1;
     2;
     4;
     8,   12;
    16,   24;
    32,   48,   72;
    64,   96,  144,  360;
   128,  192,  288,  432,  720;
   256,  384,  576,  864, 1440, 2160;
   512,  768, 1152, 1728, 2592, 2880, 4320, 10800;
  1024, 1536, 2304, 3456, 5184, 5760, 8640, 12960, 21600, 75600;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..50, flattened
• Eric Weisstein's World of Mathematics, Prime Signature
• Wikipedia, Partition (number theory)
• Wikipedia, Prime signature
• Index entries for sequences related to prime signature

Crossrefs

Column k=1-3 give: A000079, A003945 for n>2, A116453 for n>4.

Row sums give A332626.

Last elements of rows give A332644.

Cf. A000009, A087443 (for all partitions), A087980 (as sorted sequence).

Programs

Maple

b:= proc(n, i, j) option remember; `if`(i*(i+1)/2 x*ithprime(j)^i,
       b(n-i, min(n-i, i-1), j+1))[], b(n, i-1, j)[]]))
    end:
T:= n-> sort(b(n$2, 1))[]:
seq(T(n), n=0..12);

Mathematica

b[n_, i_, j_] := b[n, i, j] = If[i(i+1)/2 < n, {}, If[n == 0, {1}, Join[# * Prime[j]^i& /@ b[n - i, Min[n - i, i - 1], j + 1], b[n, i - 1, j]]]];
T[n_] := Sort[b[n, n, 1]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 07 2020, after Maple *)