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A063008 Canonical partition sequence (see A080577) encoded by prime factorization. The partition [p1,p2,p3,...] with p1 >= p2 >= p3 >= ... is encoded as 2^p1 * 3^p2 * 5^p3 * ... .

%I A063008 #36 Jun 09 2021 03:05:10
%S A063008 1,2,4,6,8,12,30,16,24,36,60,210,32,48,72,120,180,420,2310,64,96,144,
%T A063008 240,216,360,840,900,1260,4620,30030,128,192,288,480,432,720,1680,
%U A063008 1080,1800,2520,9240,6300,13860,60060,510510,256,384,576,960,864,1440,3360
%N A063008 Canonical partition sequence (see A080577) encoded by prime factorization. The partition [p1,p2,p3,...] with p1 >= p2 >= p3 >= ... is encoded as 2^p1 * 3^p2 * 5^p3 * ... .
%C A063008 Partitions are ordered first by sum. Then all partitions of n are viewed as exponent tuples on n variables and their corresponding monomials are ordered reverse lexicographically. This gives a canonical ordering: [] [1] [2,0] [1,1] [3,0,0] [2,1,0] [1,1,1] [4,0,0,0] [3,1,0,0] [2,2,0,0] [2,1,1,0] [1,1,1,1]... Rearrangement of A025487, A036035 etc.
%C A063008 Or, least integer of each prime signature; resorted in accordance with the integer partitions described in A080577. - _Alford Arnold_, Feb 13 2008
%H A063008 Alois P. Heinz, <a href="/A063008/b063008.txt">Rows n = 0..30, flattened</a>
%H A063008 S.-H. Cha, E. G. DuCasse, and L. V. Quintas, <a href="http://arxiv.org/abs/1405.5283">Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures</a>, arXiv:1405.5283 [math.NT], 2014.
%F A063008 bigomega(T(n,k)) = n. - _Andrew Howroyd_, Mar 28 2020
%e A063008 Partition [2,1,1,1] for n=5 gives 2^2*3*5*7 = 420.
%e A063008 The sequence begins:
%e A063008    1;
%e A063008    2;
%e A063008    4,  6;
%e A063008    8, 12,  30;
%e A063008   16, 24,  36,  60, 210;
%e A063008   32, 48,  72, 120, 180, 420, 2310;
%e A063008   64, 96, 144, 240, 216, 360,  840, 900, 1260, 4620, 30030;
%e A063008   ...
%p A063008 with(combinat): A063008_row := proc(n) local e,w,r;
%p A063008 r := proc(L) local B,i; B := NULL;
%p A063008 for i from nops(L) by -1 to 1 do
%p A063008 B := B,L[i] od; [%] end:
%p A063008 w := proc(e) local i, m, p, P; m := infinity;
%p A063008 P := permute([seq(ithprime(i),i=1..nops(e))]);
%p A063008 for p in P do m := min(m,mul(p[i]^e[i],i=1..nops(e))) od end:
%p A063008 [seq(w(e), e = r(partition(n)))] end:
%p A063008 seq(print(A063008_row(i)),i=0..6); # _Peter Luschny_, Jan 23 2011
%p A063008 # second Maple program:
%p A063008 b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p A063008     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A063008 T:= n-> map(x-> mul(ithprime(i)^x[i], i=1..nops(x)), b(n$2))[]:
%p A063008 seq(T(n), n=0..9);  # _Alois P. Heinz_, Sep 03 2019
%t A063008 row[n_] := Product[ Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n]; Table[row[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Dec 10 2012 *)
%t A063008 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]},Join[ Prepend[#, i]& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
%t A063008 T[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];
%t A063008 T /@ Range[0, 9] // Flatten (* _Jean-François Alcover_, Jun 09 2021, after _Alois P. Heinz_ *)
%Y A063008 Cf. A001222 (bigomega), A025487, A059901.
%Y A063008 See A080576 Maple (graded reflected lexicographic) ordering.
%Y A063008 See A080577 Mathematica (graded reverse lexicographic) ordering.
%Y A063008 See A036036 "Abramowitz and Stegun" (graded reflected colexicographic) ordering.
%Y A063008 See A036037 for graded colexicographic ordering.
%K A063008 nonn,look,tabf
%O A063008 0,2
%A A063008 Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 02 2001
%E A063008 Partially edited by _N. J. A. Sloane_, May 15, at the suggestion of _R. J. Mathar_
%E A063008 Corrected and (minor) edited by _Daniel Forgues_, Jan 03 2011