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.

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

Offset

1,2

Comments

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).

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

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

Links

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Example

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)
From Gus Wiseman, Sep 05 2018: (Start)
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)

Mathematica

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 *)

Prog

(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
-- Reinhard Zumkeller, Oct 01 2012

Crossrefs

Cf. A001055, A025487, A035310, A035341, A064553, A080444.

Cf. A007716, A056239, A162247, A215366, A275024, A317144, A317145, A318871.

Sequence in context: A278506 A255546 A358874 * A082025 A076034 A336349

Adjacent sequences: A080685 A080686 A080687 * A080689 A080690 A080691

Keyword

easy,nonn,tabf

Author

Alford Arnold, Mar 23 2003

Extensions

More terms from Sean A. Irvine, Oct 05 2011

Keyword tabf added and definition complemented accordingly by Reinhard Zumkeller, Oct 01 2012

Status

approved