The Enlightened Numbers

A Problem Proposal

Know then thyself, presume not God to scan,
The proper study of Mankind is Man.

Alexander Pope

The nature of a number
To know the prime factorization of a number is to know the "nature" of that number. Getting the prime factors is, in general, computationally expensive. Are there composite numbers n whose prime factors can simply be read off from the digits of n? Such numbers, which "know their own nature", are called "enlightened numbers". Of course, a prime number trivially exhibits all its prime factors (namely itself) in its digits, and so can be excluded from the definition.
The fundamental theorem of arithmetic attaches a central role to the prime numbers. For a number to be prime is for it to be in numerical Nirvana (the Buddhist paradise). In Buddhist canon, enlightened usually refers to someone who has understood his own nature (hence, the essential nature of things), but who has not yet entered Nirvana. Thus, an enlightened number should be thought of as a composite (non-prime) number encoding its own prime factors.
To be precise, an enlightened number is a composite number n such that the concatenation c of the prime factors of n, in increasing order, is an initial segment of n (treated as a string). Equivalently, n begins with c.
For example, the prime factors of 119911 are 11 and 991, which when concatenated yield 11991, an initial segment of 119911. Therefore, 119911 is an enlightened number. With the knowledge that 119911 is enlightened, one does not need to apply the usual time-intensive algorithms to find its prime factors. One need only read its digits from left to right, and, when one encounters a prime, check if the prime divides 119911. For instance, 11 is the first prime factor one encounters, and then 991.
There are only 31 enlightened numbers not exceeding 13436683:
250, 256, 2048, 2176, 2304, 2500, 2560, 2744, 23328, 25000, 25600, 119911, 219488, 236196, 250000, 256000, 262144, 290912, 2097152, 2238728, 2239488, 2317312, 2359296, 2370816, 2500000, 2560000, 3359232, 3515625, 3720087, 5117695, 13436683
Questions
Are there infinitely many enlightened numbers? Can you find an expression generating the enlightened numbers?
The first 31 enlightened numbers each have at most three prime factors. Is there an upper bound U to the number of prime factors of an enlightened number? If it exists, is U < 3?
Is there an enlightened number whose last digit is 9? Can the first digit of such a number be 4, 6, 7, 8, or 9? (None of the numbers in the above list ends with 9 or begins with 4, 6, 7, 8, 9.)
Some tools
For those of you who care to extend the sequence, here is Mathematica code to enumerate the enlightened numbers:
g[n_] := Map[ToString, (Transpose[FactorInteger[n]])[[1]]];
h[n_] := Module[{p, l, s, i}, p = g[n]; l = Length[p]; s = p[[1]];
Do[s = s <> p[[k]], {k, 2, l}]; s];
f[n_] := (StringPosition[ToString[n], h[n]] != {});
f2[n_] := f[n] && (StringPosition[ToString[n], g[n]])[[1]][[1]] == 1;
Do[If[! PrimeQ[n] && f2[n], Print[n]], {n, 2, 10^7}]

Here are the prime factorizations of the enlightened numbers listed above:
{{{2, 1}, {5, 3}}, {{2, 8}}, {{2, 11}}, {{2, 7}, {17, 1}}, {{2, 8}, {3, 2}}, {{2, 2}, {5, 4}}, {{2, 9}, {5, 1}}, {{2, 3}, {7, 3}}, {{2, 5}, {3, 6}}, {{2, 3}, {5, 5}}, {{2, 10}, {5, 2}}, {{11, 2}, {991, 1}}, {{2, 5}, {19, 3}}, {{2, 2}, {3, 10}}, {{2, 4}, {5, 6}}, {{2, 11}, {5, 3}}, {{2, 18}}, {{2, 5}, {9091, 1}}, {{2, 21}}, {{2, 3}, {23, 4}}, {{2, 10}, {3, 7}}, {{2, 10}, {31, 1}, {73, 1}}, {{2, 18}, {3, 2}}, {{2, 8}, {3, 3}, {7, 3}}, {{2, 5}, {5, 7}}, {{2, 12}, {5, 4}}, {{2, 9}, {3, 8}}, {{3, 2}, {5, 8}}, {{3, 12}, {7, 1}}, {{5, 1}, {11, 3}, {769, 1}}, {{13, 2}, {43, 3}}}
For example, the prime factorization of 13436683 can be read as 13² x 43³.

Joseph L. Pe

iDEN System Engineering Tools and Statistics
Motorola
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