A061394 -id:A061394 - cp's OEIS frontend

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

Michael De Vlieger, Mar 29 2019

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.

			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

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

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

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]]

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

A380146 Numbers that set records in A113901.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 48, 60, 120, 210, 240, 420, 480, 840, 1680, 3360, 6720, 13440, 26880, 36960, 53760, 73920, 107520, 147840, 215040, 295680, 591360, 960960, 1182720, 1921920, 2365440, 3843840, 4730880, 7687680, 9461760, 15375360, 30750720, 61501440, 123002880

Offset: 1

Hal M. Switkay, Jan 13 2025

nonn

All terms in this sequence are numbers of least prime signature (A025487). Therefore it is easier to search for new terms in the product of A061394 (omega of least prime signature) and A036041 (bigomega of least prime signature). Similar to but ultimately different from A378630. Terms appear to be products of primorials (A002110) with powers of 2 (A000079), and thus are never divisible by the square of an odd prime.

			omega(60) = 3 and bigomega(60) = 4; their product is 12. This product is larger than that for any smaller natural number, so 60 is a term of this sequence.

Cf. A000079, A001221, A001222, A002110, A025487, A036041, A061394, A113901, A378630.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula