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

1,4

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)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..500

Michael De Vlieger, Relations between A050327, A025487, and A002110.

Formula

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

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

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

(End)

Example

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)

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 *)

Crossrefs

Cf. A000110, A002110, A025487, A050326.

Sequence in context: A329918 A281442 A256038 * A075120 A327751 A111593

Adjacent sequences: A050324 A050325 A050326 * A050328 A050329 A050330

Keyword

nonn

Author

Christian G. Bower, Oct 15 1999

Status

approved