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%I #14 Mar 20 2016 13:14:50
%S 1,2,3,4,8,13,8,20,44,75,26,16,48,132,308,541,76,176,32,112,368,1076,
%T 2612,4683,208,604,1460,252,818,64,256,876,3408,10404,25988,47293,544,
%U 1888,5740,14300,768,2316,3172,7880,128,576,2496,10096,36848,116180
%N A tabular sequence of arrays counting ordered factorizations over least prime signatures. The unordered version is described by sequence A129306.
%C The display has 1 2 3 5 7 11 15 ... terms per column. (cf. A000041)
%C The arrays begin
%C 1.....2.....4......8......16.....32.....64......128
%C ......3.....8.....20......48....112....256......576
%C ...........13.....44.....132....368....976.....2496
%C ..................75.....308...1076...3408....10096
%C .........................541...2612..10404....36848
%C ...............................4683..25988...116180
%C .....................................47293...296564
%C .............................................545835
%C ..................26......76....208....544
%C .........................176....604...1888
%C ...............................1460...5740
%C .....................................14300
%C ................................252....768
%C ......................................2316
%C ................................818...3172
%C ......................................7880
%C with column sums
%C 1....5....25....173....1297....12225....124997 => A035341
%C Column i corresponds to partitions of i. The rows correspond successively to the partitions {i}, {i-1,1},{i-2,1,1},{i-3,1,1,1}, ..., {i-7,1,1,1,1,1,1,1}, {i-2,2}, {i-3,2,1}, {i-4,2,1,1}, {i-5,2,1,1,1}, {i-3,3}, {i-3,3,1}, {i-4,2,2}, {i-5,2,2,1}. - _Roger Lipsett_, Feb 26 2016
%e 36 = 2*2*3*3 and is in A025487. There are 26 ways to factor 36 so a(11) = 26.
%t gozinta counts ordered factorizations of an integer, and if lst is a partition we have
%t gozinta[1] = 1;
%t gozinta[n_] := gozinta[n] = 1 + Sum[gozinta[n/i], {i, Rest@Most@Divisors@n}]
%t a[lst_] := gozinta[Times @@ (Array[Prime, Length@lst]^lst)] (* _Roger Lipsett_, Feb 26 2016 *)
%Y Cf. A000041, A000670, A002033, A025487, A035098, A035341, A050324, A074206, A095705, A098348, A104725, A108464, A129306.
%K nonn
%O 1,2
%A _Alford Arnold_, Jul 10 2007
%E Corrected entries in table in comments section - _Roger Lipsett_, Feb 26 2016
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