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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A181818 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

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Last modified June 14 10:04 EDT 2021. Contains 345025 sequences. (Running on oeis4.)