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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.

%I A328524 #31 Feb 16 2025 08:33:58
%S A328524 1,2,4,8,12,16,24,32,48,72,64,96,144,360,128,192,288,432,720,256,384,
%T A328524 576,864,1440,2160,512,768,1152,1728,2592,2880,4320,10800,1024,1536,
%U A328524 2304,3456,5184,5760,8640,12960,21600,75600,2048,3072,4608,6912,10368,11520
%N 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.
%H A328524 Alois P. Heinz, <a href="/A328524/b328524.txt">Rows n = 0..50, flattened</a>
%H A328524 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSignature.html">Prime Signature</a>
%H A328524 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A328524 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_signature">Prime signature</a>
%H A328524 <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>
%e A328524 Triangle T(n,k) begins:
%e A328524      1;
%e A328524      2;
%e A328524      4;
%e A328524      8,   12;
%e A328524     16,   24;
%e A328524     32,   48,   72;
%e A328524     64,   96,  144,  360;
%e A328524    128,  192,  288,  432,  720;
%e A328524    256,  384,  576,  864, 1440, 2160;
%e A328524    512,  768, 1152, 1728, 2592, 2880, 4320, 10800;
%e A328524   1024, 1536, 2304, 3456, 5184, 5760, 8640, 12960, 21600, 75600;
%e A328524   ...
%p A328524 b:= proc(n, i, j) option remember; `if`(i*(i+1)/2<n, [],
%p A328524       `if`(n=0, [1], [map(x-> x*ithprime(j)^i,
%p A328524        b(n-i, min(n-i, i-1), j+1))[], b(n, i-1, j)[]]))
%p A328524     end:
%p A328524 T:= n-> sort(b(n$2, 1))[]:
%p A328524 seq(T(n), n=0..12);
%t A328524 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 A328524 T[n_] := Sort[b[n, n, 1]];
%t A328524 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, May 07 2020, after Maple *)
%Y A328524 Column k=1-3 give: A000079, A003945 for n>2, A116453 for n>4.
%Y A328524 Row sums give A332626.
%Y A328524 Last elements of rows give A332644.
%Y A328524 Cf. A000009, A087443 (for all partitions), A087980 (as sorted sequence).
%K A328524 nonn,tabf
%O A328524 0,2
%A A328524 _Alois P. Heinz_, Feb 18 2020