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A050327
Number of factorizations into distinct squarefree factors indexed by prime signatures. A050326(A025487).
2
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)
FORMULA
From Michael De Vlieger, Oct 06 2017: (Start)
a(n) = A050326(A025487(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
KEYWORD
nonn
AUTHOR
Christian G. Bower, Oct 15 1999
STATUS
approved