A373326 a(1) = 1, a(2) = 2. For n > 2, let i = a(n-2), j = a(n-1). Then a(n) is least novel k such that A007947(i*j*k) is the smallest possible primorial number, subject to no more than two consecutive terms having the same number of distinct prime divisors.

1, 2, 3, 6, 4, 8, 12, 9, 16, 18, 24, 27, 32, 36, 48, 64, 54, 72, 81, 96, 108, 128, 144, 162, 243, 192, 216, 256, 288, 324, 512, 384, 432, 729, 486, 576, 1024, 648

Offset

1,2

Comments

For all consecutive i,j,k rad(i*j*k) = A002110(2) = 6.

If A001221(i,j) = omega(i,j) = {1,1}, omega(k) is constrained to 2, if omega(i,j) = {2,2}, omega(k) must = 1. If omega(i,j) = {1,2} or {2,1} then omega(k) is 1 or 2, depending on smallest missing number conforming to the definition.

The sequence is a greedy permutation of the 3-smooth numbers (A003586). A power of 2 or 3 follows any pair of consecutive terms each having two distinct prime factors, but the converse is not true (8,12 --> 9 and 12,9 -->16) for example.

Links

Table of n, a(n) for n=1..38.

Example

a(1,2) = 1,2 with omega(1,2) = 0,1 so a(3) = 3 since rad(1*2*3) = 6, the omega condition is satisfied (omega(3) = 1) and 3 is least such term.
a(2,3) = 2,3, omega(2,3) = 1,1 so a(4) cannot be 4 or 5 since both have omega = 1, even though rad(2*3*4) and rad(2*3*5) are both primorial. Therefore a(4) = 6, omega(6) = 2, and rad(2*3*6) = 6.
a(5) = 4 because rad(3*6*4) = 6, omega(3,6,4) = 1,2,1 and 4 is least such term.

Crossrefs

Cf. A001221, A002110, A007947.

Sequence in context: A289055 A109890 A370046 * A086537 A212486 A127562

Adjacent sequences: A373323 A373324 A373325 * A373327 A373328 A373329

Keyword

nonn,more,new

Author

David James Sycamore, Jun 01 2024

Status

approved