OFFSET
0,2
COMMENTS
For the lexicographic ordering, the prime factors must be written in nonincreasing order. If we write the factors in nondecreasing order, we get a lexicographically ordered set with an order type that is greater than a natural number index - the resulting sequence does not include all qualifying numbers. (Note also that the symbols used for the lexicographic order are the prime numbers, not their digits.)
a(n) is the n-th power of 4 in the monoid defined in A331590.
Conjecture: a(n) is the position of the first occurrence of n in A334109.
FORMULA
EXAMPLE
The initial terms are shown below, equated with the product of their prime factors to exhibit the lexicographic ordering. The list starts with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
1 = .
4 = 2*2.
9 = 3*3.
36 = 3*3*2*2.
25 = 5*5.
100 = 5*5*2*2.
225 = 5*5*3*3.
900 = 5*5*3*3*2*2.
49 = 7*7.
196 = 7*7*2*2.
441 = 7*7*3*3.
MATHEMATICA
Array[If[# == 0, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[3^#]]] &, 51, 0] (* Michael De Vlieger, May 26 2020 *)
PROG
(PARI) A334110(n) = { my(p=2, m=1); while(n, if(n%2, m *= p^2); n >>= 1; p = nextprime(1+p)); (m); };
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen and Peter Munn, May 01 2020
STATUS
approved