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A080688
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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.
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11
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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, 16, 20, 25, 34, 47, 53, 18, 33, 38, 59, 61, 28, 35, 46, 67, 39, 71, 58, 73, 79, 24, 30, 44, 51, 55, 62, 83, 49, 89, 74, 97, 27, 57, 101, 52, 65, 82
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OFFSET
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1,2
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COMMENTS
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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).
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
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LINKS
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EXAMPLE
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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)
Triangle begins:
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
16 20 25 34 47
Corresponding triangle of factorizations begins:
(),
(2),
(3),
(2*2), (4),
(5),
(2*3), (6),
(7),
(2*2*2), (2*4), (8),
(3*3), (9),
(2*5), (10),
(11),
(2*2*3), (3*4), (2*6), (12).
(End)
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];
Table[Sort[Table[Times@@Prime/@(f-1), {f, facs[n]}]], {n, 20}] (* Gus Wiseman, Sep 05 2018 *)
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PROG
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(Haskell)
a080688 n k = a080688_row n !! (k-1)
a080688_row n = map (+ 1) $ take (a001055 n) $
elemIndices n $ map fromInteger a064553_list
a080688_tabl = map a080688_row [1..]
a080688_list = concat a080688_tabl
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CROSSREFS
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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EXTENSIONS
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Keyword tabf added and definition complemented accordingly by Reinhard Zumkeller, Oct 01 2012
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STATUS
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approved
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