A307107 a(n) = A025487(n)/A247451(n).

1, 1, 2, 1, 4, 2, 8, 4, 1, 16, 6, 8, 2, 32, 12, 16, 4, 64, 24, 6, 32, 1, 36, 8, 128, 48, 12, 64, 2, 72, 16, 256, 96, 24, 128, 4, 144, 30, 32, 512, 36, 192, 6, 216, 48, 256, 8, 288, 60, 64, 1024, 72, 384, 1, 12, 432, 96, 512, 16, 576, 120, 128, 2048, 144, 768, 2

Offset

1,3

Comments

Ratio of A025487(n) and the largest primorial that divides A025487(n). The largest primorial that divides A025487(n) is A002110(A001221(n)).

If A025487(n) is a primorial (i.e., in A002110), a(n) = 1.

a(n) is in A025487 by definition of that sequence as a sorted list of products of primorials.

Conjectures:

1. 1 is the most common value in this sequence even though it only pertains to primorials.

2. All terms in A025487 are in this sequence.

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Formula

a(n) = A025487(n)/A002110(A061394(n)) = A025487(n)/A002110(A001221(A025487(n))).

Example

We can represent the prime divisors p with multiplicity of A025487(n) in a chart where the columns pertain to p and the rows multiplicity. In such a chart, A247451(n) is the longest row (marked by "O" below), and a(n) is the product of primes left over (marked by "X") when we eliminate the primes that produce A247451(n).
A025487(9) =       30 = a(9) * A247451(9)
                      = 1 * 30
  1  O O O
     2 3 5
A025487(27) =     360 = a(27) * A247451(27)
                      = 12 * 30
  3  X
  2  X X
  1  O O O
     2 3 5
A025487(183) = 166320 = a(183) * A247451(183)
                      = 72 * 2310
  4  X
  3  X X
  2  X X
  1  O O O O O
     2 3 5 7 11

Mathematica

f[n_] := {{1}}~Join~Block[{lim = Product[Prime@ i, {i, n}], ww = NestList[Append[#, 1] &, {1}, n - 1], g}, g[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]]; Map[Block[{w = #, k = 1}, Sort@ Prepend[If[Length@ # == 0, #, #[[1]]], Product[Prime@ i, {i, Length@ w}]] &@ Reap[Do[If[# < lim, Sow[#]; k = 1, If[k >= Length@ w, Break[], k++]] &@ g@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@#] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1]]], {i, Infinity}]][[-1]]] &, ww]]; With[{s = Union@ Flatten@ f@ 6}, Map[#/Product[Prime@ i, {i, PrimeNu@ #}] &, s]]

Crossrefs

Cf. A001221, A002110, A025487, A061394, A247451.

Sequence in context: A117000 A082392 A233327 * A085086 A274623 A265256

Adjacent sequences: A307104 A307105 A307106 * A307108 A307109 A307110

Keyword

nonn,look

Author

Michael De Vlieger, Mar 29 2019

Status

approved