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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. 5
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
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, 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
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Feb 18 2020
STATUS
approved

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Last modified April 15 10:24 EDT 2024. Contains 371681 sequences. (Running on oeis4.)