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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 * ... . 33
 1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 240, 216, 360, 840, 900, 1260, 4620, 30030, 128, 192, 288, 480, 432, 720, 1680, 1080, 1800, 2520, 9240, 6300, 13860, 60060, 510510, 256, 384, 576, 960, 864, 1440, 3360 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. Or, least integer of each prime signature; resorted in accordance with the integer partitions described in A080577. - Alford Arnold, Feb 13 2008 LINKS Alois P. Heinz, Rows n = 0..30, flattened S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014. FORMULA bigomega(T(n,k)) = n. - Andrew Howroyd, Mar 28 2020 EXAMPLE Partition [2,1,1,1] for n=5 gives 2^2*3*5*7 = 420. The sequence begins:    1;    2;    4,  6;    8, 12,  30;   16, 24,  36,  60, 210;   32, 48,  72, 120, 180, 420, 2310;   64, 96, 144, 240, 216, 360,  840, 900, 1260, 4620, 30030;   ... MAPLE with(combinat): A063008_row := proc(n) local e, w, r; r := proc(L) local B, i; B := NULL; for i from nops(L) by -1 to 1 do B := B, L[i] od; [%] end: w := proc(e) local i, m, p, P; m := infinity; P := permute([seq(ithprime(i), i=1..nops(e))]); for p in P do m := min(m, mul(p[i]^e[i], i=1..nops(e))) od end: [seq(w(e), e = r(partition(n)))] end: seq(print(A063008_row(i)), i=0..6); # Peter Luschny, Jan 23 2011 # second Maple program: b:= (n, i)-> `if`(n=0 or i=1, [[1\$n]], [map(x->     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]): T:= n-> map(x-> mul(ithprime(i)^x[i], i=1..nops(x)), b(n\$2))[]: seq(T(n), n=0..9);  # Alois P. Heinz, Sep 03 2019 MATHEMATICA row[n_] := Product[ Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n]; Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 10 2012 *) CROSSREFS Cf. A001222 (bigomega), A025487, A059901. See A080576 Maple (graded reflected lexicographic) ordering. See A080577 Mathematica (graded reverse lexicographic) ordering. See A036036 "Abramowitz and Stegun" (graded reflected colexicographic) ordering. See A036037 for graded colexicographic ordering. Sequence in context: A194406 A087443 A036035 * A059901 A303555 A136101 Adjacent sequences:  A063005 A063006 A063007 * A063009 A063010 A063011 KEYWORD nonn,look,tabf AUTHOR Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 02 2001 EXTENSIONS Partially edited by N. J. A. Sloane, May 15, at the suggestion of R. J. Mathar Corrected and (minor) edited by Daniel Forgues, Jan 03 2011 STATUS approved

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Last modified April 18 02:38 EDT 2021. Contains 343072 sequences. (Running on oeis4.)