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%I #16 Sep 11 2018 05:03:06
%S 1,2,3,4,5,7,6,11,13,8,10,17,9,19,14,23,29,12,15,22,31,37,26,41,21,43,
%T 16,20,25,34,47,53,18,33,38,59,61,28,35,46,67,39,71,58,73,79,24,30,44,
%U 51,55,62,83,49,89,74,97,27,57,101,52,65,82
%N Resort the index of A064553 using A080444 and maintaining ascending order within each grouping: seen as a triangle read by rows, the n-th row contains the A001055(n) numbers m with A064553(m)=n.
%C The number 12 can be written as 3*2*2, 4*3, 6*2 and 12 corresponding to each of the four values (12,15,22,31) in the example. Note that A001055(12) = 4. Since A001055(n) depends only on the least prime signature, the values 1,2,4,6,8,12,16,24,30,32,36,... A025487 are of special interest when counting multisets. (see for example, A035310 and a035341).
%C A064553(T(n,k)) = A080444(n,k) = n for k=1..A001055(n); T(n,1) = A064554(n); T(n,A001055(n)) = A064554(n). - _Reinhard Zumkeller_, Oct 01 2012
%C Row n is the sorted list of shifted Heinz numbers of factorizations of n into factors > 1, where the shifted Heinz number of a factorization (y_1, ..., y_k) is prime(y_1 - 1) * ... * prime(y_k - 1). - _Gus Wiseman_, Sep 05 2018
%H Reinhard Zumkeller, <a href="/A080688/b080688.txt">Rows n = 1..1000 of triangle, flattened</a>
%e a(18),a(19),a(20) and a(21) are 12,15,22 and 31 because A064553(12,15,22,31) = (12,12,12,12) similarly, A064553(36,45,66,76,93,95,118,121,149) = (36,36,36,36,36,36,36,36,36)
%e From _Gus Wiseman_, Sep 05 2018: (Start)
%e Triangle begins:
%e 1
%e 2
%e 3
%e 4 5
%e 7
%e 6 11
%e 13
%e 8 10 17
%e 9 19
%e 14 23
%e 29
%e 12 15 22 31
%e 37
%e 26 41
%e 21 43
%e 16 20 25 34 47
%e Corresponding triangle of factorizations begins:
%e (),
%e (2),
%e (3),
%e (2*2), (4),
%e (5),
%e (2*3), (6),
%e (7),
%e (2*2*2), (2*4), (8),
%e (3*3), (9),
%e (2*5), (10),
%e (11),
%e (2*2*3), (3*4), (2*6), (12).
%e (End)
%t facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
%t Table[Sort[Table[Times@@Prime/@(f-1),{f,facs[n]}]],{n,20}] (* _Gus Wiseman_, Sep 05 2018 *)
%o (Haskell)
%o a080688 n k = a080688_row n !! (k-1)
%o a080688_row n = map (+ 1) $ take (a001055 n) $
%o elemIndices n $ map fromInteger a064553_list
%o a080688_tabl = map a080688_row [1..]
%o a080688_list = concat a080688_tabl
%o -- _Reinhard Zumkeller_, Oct 01 2012
%Y Cf. A001055, A025487, A035310, A035341, A064553, A080444.
%Y Cf. A007716, A056239, A162247, A215366, A275024, A317144, A317145, A318871.
%K easy,nonn,tabf
%O 1,2
%A _Alford Arnold_, Mar 23 2003
%E More terms from _Sean A. Irvine_, Oct 05 2011
%E Keyword tabf added and definition complemented accordingly by _Reinhard Zumkeller_, Oct 01 2012
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