A260850 Lexicographically earliest sequence such that for any n>1, n=u*v, where u/v = a(n)/a(n-1) in reduced form.

1, 2, 6, 24, 120, 20, 140, 1120, 10080, 1008, 11088, 924, 12012, 858, 1430, 22880, 388960, 1750320, 33256080, 1662804, 3879876, 176358, 4056234, 10816624, 270415600, 10400600, 280816200, 10029150, 290845350, 9694845, 300540195, 9617286240, 35263382880

Offset

1,2

Comments

a(n) is obtained from a(n - 1) by multiplying by the product of all prime powers p^e of n whose exponent e exceeds the exponent of p in a(n - 1), and dividing by the product of the remaining prime powers of n. - Felix Huber, May 27 2026

Links

Paul Tek, Table of n, a(n) for n = 1..3365

Paul Tek, PARI program for this sequence

Michael De Vlieger, Plot p(i)^m(i) | a(n) at (x,y) = (n,i), n = 1..2048, 3X vertical exaggeration, with a color function showing m(i) = 1 in black, m(i) = 2 in red, ..., largest m(i) in the dataset in magenta.

Michael De Vlieger, Prime Power Decomposition of a(n), n = 1..1000.

Formula

a(p) = p*a(p-1) for any prime p.

a(n) = A008336(n+1) for n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 21, 22, 23; are there other indices with this property?

a(1) = 1. For n > 1, write n = Product_{i = 1 .. k} p_i^e_i. Then a(n) = a(n - 1)*Product_{i = 1 .. k} p_i^{v_i*e_i}, where v_i = +1 if e_i > v_{p_i} (a(n - 1)), and v_i = -1 otherwise. - Felix Huber, May 27 2026

Example

From Michael De Vlieger, Apr 12 2024: (Start)
Table showing exponents m of prime powers p^m | a(n), n = 1..20, with "." representing p < gpf(n) does not divide a(n):
                       1111
    n        a(n)  23571379
   ------------------------
    1          1   .
    2          2   1
    3          6   11
    4         24   31
    5        120   311
    6         20   2.1
    7        140   2.11
    8       1120   5.11
    9      10080   5211
   10       1008   42.1
   11      11088   42.11
   12        924   21.11
   13      12012   21.111
   14        858   11..11
   15       1430   1.1.11
   16      22880   5.1.11
   17     388960   5.1.111
   18    1750320   421.111
   19   33256080   421.1111
   20    1662804   22..1111 (End)

Maple

A260850List := proc(N)
    local a, b, i, n, p;
    a := Vector(1 .. N);
    b := 1;
    for n from 1 to N do
        for i in ifactors(n)[2] do
            p := i[1]^i[2];
            b := `if`(b mod p = 0, b/p, b*p);
        end do;
        a[n] := b;
    end do;
    convert(a, list);
end proc:
A260850List(100); # Felix Huber, May 27 2026

Mathematica

nn = 35; p[_] := 0; r = 0;
Do[(Map[If[p[#1] < #2,
      p[#1] += #2,
      p[#1] -= #2] & @@ # &, #];
      If[r < #, r = #] &[#[[-1, 1]] ] ) &@
    Map[{PrimePi[#1], #2} & @@ # &, FactorInteger[n]];
  a[n] = Times @@ Array[Prime[#]^p[#] &, r], {n, nn}];
Array[a, nn] (* Michael De Vlieger, Apr 12 2024 *)

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A008336, A370974 (sorted version).

Sequence in context: A284567 A360300 A065422 * A008336 A360298 A033643

Adjacent sequences: A260847 A260848 A260849 * A260851 A260852 A260853

Keyword

nonn

Author

Paul Tek, Aug 01 2015

Status

approved