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%I #75 Apr 06 2019 01:18:01
%S 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,
%T 2,72,16,256,96,24,128,4,144,30,32,512,36,192,6,216,48,256,8,288,60,
%U 64,1024,72,384,1,12,432,96,512,16,576,120,128,2048,144,768,2
%N a(n) = A025487(n)/A247451(n).
%C Ratio of A025487(n) and the largest primorial that divides A025487(n). The largest primorial that divides A025487(n) is A002110(A001221(n)).
%C If A025487(n) is a primorial (i.e., in A002110), a(n) = 1.
%C a(n) is in A025487 by definition of that sequence as a sorted list of products of primorials.
%C Conjectures:
%C 1. 1 is the most common value in this sequence even though it only pertains to primorials.
%C 2. All terms in A025487 are in this sequence.
%H David A. Corneth, <a href="/A307107/b307107.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A025487(n)/A002110(A061394(n)) = A025487(n)/A002110(A001221(A025487(n))).
%e 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).
%e A025487(9) = 30 = a(9) * A247451(9)
%e = 1 * 30
%e 1 O O O
%e 2 3 5
%e A025487(27) = 360 = a(27) * A247451(27)
%e = 12 * 30
%e 3 X
%e 2 X X
%e 1 O O O
%e 2 3 5
%e A025487(183) = 166320 = a(183) * A247451(183)
%e = 72 * 2310
%e 4 X
%e 3 X X
%e 2 X X
%e 1 O O O O O
%e 2 3 5 7 11
%t 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]]
%Y Cf. A001221, A002110, A025487, A061394, A247451.
%K nonn,look
%O 1,3
%A _Michael De Vlieger_, Mar 29 2019
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