A063008 - OEIS

%I #36 Jun 09 2021 03:05:10

%S 1,2,4,6,8,12,30,16,24,36,60,210,32,48,72,120,180,420,2310,64,96,144,

%T 240,216,360,840,900,1260,4620,30030,128,192,288,480,432,720,1680,

%U 1080,1800,2520,9240,6300,13860,60060,510510,256,384,576,960,864,1440,3360

%N 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 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 Or, least integer of each prime signature; resorted in accordance with the integer partitions described in A080577. - _Alford Arnold_, Feb 13 2008

%H Alois P. Heinz, <a href="/A063008/b063008.txt">Rows n = 0..30, flattened</a>

%H 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 bigomega(T(n,k)) = n. - _Andrew Howroyd_, Mar 28 2020

%e Partition [2,1,1,1] for n=5 gives 2^2*3*5*7 = 420.

%e The sequence begins:

%e    1;

%e    2;

%e    4,  6;

%e    8, 12,  30;

%e   16, 24,  36,  60, 210;

%e   32, 48,  72, 120, 180, 420, 2310;

%e   64, 96, 144, 240, 216, 360,  840, 900, 1260, 4620, 30030;

%e   ...

%p with(combinat): A063008_row := proc(n) local e,w,r;

%p r := proc(L) local B,i; B := NULL;

%p for i from nops(L) by -1 to 1 do

%p B := B,L[i] od; [%] end:

%p w := proc(e) local i, m, p, P; m := infinity;

%p P := permute([seq(ithprime(i),i=1..nops(e))]);

%p for p in P do m := min(m,mul(p[i]^e[i],i=1..nops(e))) od end:

%p [seq(w(e), e = r(partition(n)))] end:

%p seq(print(A063008_row(i)),i=0..6); # _Peter Luschny_, Jan 23 2011

%p # second Maple program:

%p b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->

%p     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):

%p T:= n-> map(x-> mul(ithprime(i)^x[i], i=1..nops(x)), b(n$2))[]:

%p seq(T(n), n=0..9);  # _Alois P. Heinz_, Sep 03 2019

%t 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 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 T[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];

%t T /@ Range[0, 9] // Flatten (* _Jean-François Alcover_, Jun 09 2021, after _Alois P. Heinz_ *)

%Y Cf. A001222 (bigomega), A025487, A059901.

%Y See A080576 Maple (graded reflected lexicographic) ordering.

%Y See A080577 Mathematica (graded reverse lexicographic) ordering.

%Y See A036036 "Abramowitz and Stegun" (graded reflected colexicographic) ordering.

%Y See A036037 for graded colexicographic ordering.

%K nonn,look,tabf

%O 0,2

%A Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 02 2001

%E Partially edited by _N. J. A. Sloane_, May 15, at the suggestion of _R. J. Mathar_

%E Corrected and (minor) edited by _Daniel Forgues_, Jan 03 2011