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

0,2

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

Example

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;
  ...

Maple

b:= proc(n, i, j) option remember; `if`(i*(i+1)/2<n, [],
      `if`(n=0, [1], [map(x-> 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 *)

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).

Sequence in context: A363063 A336496 A317804 * A322447 A170892 A246468

Adjacent sequences: A328521 A328522 A328523 * A328525 A328526 A328527

Keyword

nonn,tabf

Author

Alois P. Heinz, Feb 18 2020

Status

approved