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]]

Prog

(Python)
from math import prod
from itertools import count
from functools import lru_cache
from sympy import prime, integer_log, primefactors
from oeis_sequences.OEISsequences import bisection
def A307107(n):
    @lru_cache(maxsize=None)
    def g(x, m, j): return sum(g(x//(prime(m)**i), m-1, i) for i in range(j, integer_log(x, prime(m))[0]+1)) if m-1 else max(0, x.bit_length()-j)
    def f(x):
        c, p = n-1+x, 1
        for k in count(1):
            p *= prime(k)
            if p>x:
                break
            c -= g(x, k, 1)
        return c
    return (m:=bisection(f, n, n))//prod(primefactors(m)) # Chai Wah Wu, Mar 26 2026

Crossrefs

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

Sequence in context: A082392 A377377 A233327 * A085086 A274623 A265256

Adjacent sequences: A307104 A307105 A307106 * A307108 A307109 A307110

Keyword

nonn,look

Author

Michael De Vlieger, Mar 29 2019

Status

approved