A299442 Lexicographically earliest sequence of distinct positive terms such that, for any n > 1, if prime(k) is the greatest prime factor of a(n) then k divides a(n+1) (where prime(k) denotes the k-th prime).

1, 2, 3, 4, 5, 6, 8, 7, 12, 10, 9, 14, 16, 11, 15, 18, 20, 21, 24, 22, 25, 27, 26, 30, 33, 35, 28, 32, 13, 36, 34, 42, 40, 39, 48, 38, 56, 44, 45, 51, 49, 52, 54, 46, 63, 60, 57, 64, 17, 70, 68, 77, 50, 66, 55, 65, 72, 58, 80, 69, 81, 62, 88, 75, 78, 84, 76

Offset

1,2

Comments

In other words, for any n > 1, A061395(a(n)) divides a(n+1).

See also A299441 for the variant involving least prime factors.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A299442

Rémy Sigrist, Colored scatterplot of the first 100000 terms (where the color is function of A006530(a(n-1)))

Example

The first terms, alongside A061395(a(n)), are:
  n     a(n)    A061395(a(n))
  --    ----    -------------
   1       1       0
   2       2       1
   3       3       2
   4       4       1
   5       5       3
   6       6       2
   7       8       1
   8       7       4
   9      12       2
  10      10       3
  11       9       2
  12      14       4
  13      16       1
  14      11       5
  15      15       3
  16      18       2
  17      20       3
  18      21       4
  19      24       2
  20      22       5

Maple

N:= 1000: # to get terms before the first term > N
with(numtheory):
V:= Vector(N):
A[1]:= 1: A[2]:= 2: V[1]:= 1: V[2]:= 1:
found:= true:
for n from 2 while found do
     found:= false;
     k:= pi(max(factorset(A[n])));
     for v from k to N by k do
       if V[v] = 0 then
         V[v]:= 1;
         A[n+1]:= v;
         found:= true;
         break
       fi
     od
od:
seq(A[i], i=1..n-1); # Robert Israel, Feb 18 2018

Mathematica

max = 100; Clear[a, V]; a[_] = 0; V[_] = 0; a[1] = 1; a[2] = 2; V[1] = 1; V[2] = 1; found = True; For[n = 2, found, n++, found = False; k = PrimePi[ FactorInteger[a[n]][[-1, 1]]]; For[v = k, v <= max, v += k, If[V[v] == 0, V[v] = 1; a[n+1] = v; found = True; Break[]]]]; DeleteCases[ Array[a, max], 0] (* Jean-François Alcover, Feb 23 2018, after Robert Israel *)

Prog

(PARI) See Links section.

Crossrefs

Cf. A006530, A061395, A299441.

Sequence in context: A274607 A339607 A262374 * A299440 A330401 A145518

Adjacent sequences: A299439 A299440 A299441 * A299443 A299444 A299445

Keyword

nonn

Author

Rémy Sigrist, Feb 10 2018

Status

approved