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
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