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A080688 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.

%I A080688 #19 Jul 12 2024 21:54:24
%S A080688 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 A080688 16,20,25,34,47,53,18,33,38,59,61,28,35,46,67,39,71,58,73,79,24,30,44,
%U A080688 51,55,62,83,49,89,74,97,27,57,101,52,65,82
%N A080688 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 A080688 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 A080688 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 A080688 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 A080688 Reinhard Zumkeller, <a href="/A080688/b080688.txt">Rows n = 1..1000 of triangle, flattened</a>
%e A080688 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 A080688 From _Gus Wiseman_, Sep 05 2018: (Start)
%e A080688 Triangle begins:
%e A080688    1
%e A080688    2
%e A080688    3
%e A080688    4  5
%e A080688    7
%e A080688    6 11
%e A080688   13
%e A080688    8 10 17
%e A080688    9 19
%e A080688   14 23
%e A080688   29
%e A080688   12 15 22 31
%e A080688   37
%e A080688   26 41
%e A080688   21 43
%e A080688   16 20 25 34 47
%e A080688 Corresponding triangle of factorizations begins:
%e A080688   (),
%e A080688   (2),
%e A080688   (3),
%e A080688   (2*2), (4),
%e A080688   (5),
%e A080688   (2*3), (6),
%e A080688   (7),
%e A080688   (2*2*2), (2*4), (8),
%e A080688   (3*3), (9),
%e A080688   (2*5), (10),
%e A080688   (11),
%e A080688   (2*2*3), (3*4), (2*6), (12).
%e A080688 (End)
%t A080688 facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
%t A080688 Table[Sort[Table[Times@@Prime/@(f-1),{f,facs[n]}]],{n,20}] (* _Gus Wiseman_, Sep 05 2018 *)
%o A080688 (Haskell)
%o A080688 a080688 n k = a080688_row n !! (k-1)
%o A080688 a080688_row n = map (+ 1) $ take (a001055 n) $
%o A080688                 elemIndices n $ map fromInteger a064553_list
%o A080688 a080688_tabl = map a080688_row [1..]
%o A080688 a080688_list = concat a080688_tabl
%o A080688 -- _Reinhard Zumkeller_, Oct 01 2012
%Y A080688 Cf. A001055, A025487, A035310, A035341, A064553, A080444.
%Y A080688 Cf. A007716, A056239, A162247, A215366, A275024, A317144, A317145, A318871.
%K A080688 easy,nonn,tabf
%O A080688 1,2
%A A080688 _Alford Arnold_, Mar 23 2003
%E A080688 More terms from _Sean A. Irvine_, Oct 05 2011
%E A080688 Keyword tabf added and definition complemented accordingly by _Reinhard Zumkeller_, Oct 01 2012