123 is a peculiar integer, because 1+2+3=1*2*3. 1412 is also peculiar, since 1+4+1+2=1*4*1*2.

A simple question: are there infinitely many such numbers?

A not so simple question: if so, are there such numbers for ANY number of digits?

Pick a list of N digits between 2 and 9, and call it D. Let S be their sum, and P their product. The number formed by (P-S) ones, followed by the digits in D, is "peculiar". (And so are all the permutations of its digits.) The number has N+P-S digits.

For the second part, the question is if EVERY number from 1 onwards can always be written as N+P-S... my bet is on "NO", just because I think finding a counterexample might be easier than finding a proof!