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?

It seems obvious that there are infinitely many such numbers:
Take any product of integers 2-9 [2*2*3 = 12] its sum [2+2+3 = 7] will be less than this.  So you tack on ones to make up this difference [11111223 is then another number with this property].

Can it be done with any number of digits?  I'm not sure but the key is the difference between the product and the sum minus the number of digits used.

Two digits: 22
Three digits: 123
Four digits: 1124
Five digits: 11125 or 11133
...
Ten digits: 1111111144
Eleven digits: 11111111136

They get harder to find, but since there are more possibilities there are probably more examples for each number of digits.  I see no reason why there should not be numbers of any given length.

Posted by Jer on 2006-08-21 10:03:34