A050327 Number of factorizations into distinct squarefree factors indexed by prime signatures. A050326(A025487).

1, 1, 0, 2, 0, 1, 0, 0, 5, 0, 1, 0, 4, 0, 0, 0, 1, 0, 0, 5, 0, 15, 0, 0, 0, 0, 2, 0, 16, 0, 0, 0, 0, 0, 0, 7, 0, 8, 0, 0, 1, 0, 23, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 52, 14, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 68, 3, 0, 4, 0, 0, 40, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 41

Offset: 1

Author

Christian G. Bower, Oct 15 1999

Keywords

• nonn

Comments

From Michael De Vlieger, Oct 06 2017: (Start)

Terms in A025487 that set records in this sequence: {1, 6, 30, 210, 420, 1260, 2310, 4620, 13860, 30030, 60060, 180180, 510510, 900900, 1021020, 3063060, 6126120, 9699690, ...}.

Conjecture: prime signatures corresponding to primorials A002110(i) with i > 1 set records in this sequence. (End)

Examples

			From _Michael De Vlieger_, Oct 06 2017: (Start)
First 20 values, with numbers in column "r" records, and the last column the concatenation of exponents of standard form prime decomposition of A025487(n):
  .
   n    a(n)   r   A025487(n) rev(A054841(A025487(n)))
  --------------------------------------------
   1      1    1          1   0
   2      1               2   1
   3      0               4   2
   4      2    2          6   11
   5      0               8   3
   6      1              12   21
   7      0              16   4
   8      0              24   31
   9      5    3         30   111
  10      0              32   5
  11      1              36   22
  12      0              48   41
  13      4              60   211
  14      0              64   6
  15      0              72   32
  16      0              96   51
  17      1             120   311
  18      0             128   7
  19      0             144   42
  20      5             180   221
  21      0             192   61
  22     15    4        210   1111
(End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..500
• Michael De Vlieger, Relations between A050327, A025487, and A002110.

Crossrefs

Cf. A000110, A002110, A025487, A050326.

Programs

Mathematica

f[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[f[n/d], Min @@ # > d &]], {d, Select[Rest@ Divisors@ n, SquareFreeQ]}]]; Length[f@ #] & /@ Prepend[#, 1] &@ Sort@ Map[Times @@ Flatten@ MapIndexed[ConstantArray[Prime@ First@ #2, #1] &, #] &, Union@ Table[Sort[FactorInteger[n][[All, -1]], Greater], {n, 2, Product[Prime@ i, {i, 7}]}]] (* Michael De Vlieger, Oct 06 2017, after Gus Wiseman at A293243 *)

Formula

From Michael De Vlieger, Oct 06 2017: (Start)

a(n) = A050326(A025487(n)).

A050326(A002110(n)) = A000110(n).

(End)