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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.
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%I #30 May 07 2020 08:01:51

%S 1,2,4,8,12,16,24,32,48,72,64,96,144,360,128,192,288,432,720,256,384,

%T 576,864,1440,2160,512,768,1152,1728,2592,2880,4320,10800,1024,1536,

%U 2304,3456,5184,5760,8640,12960,21600,75600,2048,3072,4608,6912,10368,11520

%N 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.

%H Alois P. Heinz, <a href="/A328524/b328524.txt">Rows n = 0..50, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSignature.html">Prime Signature</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_signature">Prime signature</a>

%H <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>

%e Triangle T(n,k) begins:

%e 1;

%e 2;

%e 4;

%e 8, 12;

%e 16, 24;

%e 32, 48, 72;

%e 64, 96, 144, 360;

%e 128, 192, 288, 432, 720;

%e 256, 384, 576, 864, 1440, 2160;

%e 512, 768, 1152, 1728, 2592, 2880, 4320, 10800;

%e 1024, 1536, 2304, 3456, 5184, 5760, 8640, 12960, 21600, 75600;

%e ...

%p b:= proc(n, i, j) option remember; `if`(i*(i+1)/2<n, [],

%p `if`(n=0, [1], [map(x-> x*ithprime(j)^i,

%p b(n-i, min(n-i, i-1), j+1))[], b(n, i-1, j)[]]))

%p end:

%p T:= n-> sort(b(n$2, 1))[]:

%p seq(T(n), n=0..12);

%t 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 T[n_] := Sort[b[n, n, 1]];

%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, May 07 2020, after Maple *)

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

%Y Row sums give A332626.

%Y Last elements of rows give A332644.

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

%K nonn,tabf

%O 0,2

%A _Alois P. Heinz_, Feb 18 2020