A050327 - OEIS

%I #15 Oct 09 2017 11:12:29

%S 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,

%T 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,

%U 68,3,0,4,0,0,40,0,0,0,0,11,0,0,0,0,0,41

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

%C From _Michael De Vlieger_, Oct 06 2017: (Start)

%C 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, ...}.

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

%H Michael De Vlieger, <a href="/A050327/b050327.txt">Table of n, a(n) for n = 1..500</a>

%H Michael De Vlieger, <a href="/A050327/a050327.txt">Relations between A050327, A025487, and A002110.</a>

%F From _Michael De Vlieger_, Oct 06 2017: (Start)

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

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

%F (End)

%e From _Michael De Vlieger_, Oct 06 2017: (Start)

%e 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):

%e   .

%e    n    a(n)   r   A025487(n) rev(A054841(A025487(n)))

%e   --------------------------------------------

%e    1      1    1          1   0

%e    2      1               2   1

%e    3      0               4   2

%e    4      2    2          6   11

%e    5      0               8   3

%e    6      1              12   21

%e    7      0              16   4

%e    8      0              24   31

%e    9      5    3         30   111

%e   10      0              32   5

%e   11      1              36   22

%e   12      0              48   41

%e   13      4              60   211

%e   14      0              64   6

%e   15      0              72   32

%e   16      0              96   51

%e   17      1             120   311

%e   18      0             128   7

%e   19      0             144   42

%e   20      5             180   221

%e   21      0             192   61

%e   22     15    4        210   1111

%e (End)

%t 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 *)

%Y Cf. A000110, A002110, A025487, A050326.

%K nonn

%O 1,4

%A _Christian G. Bower_, Oct 15 1999