[seqfan] Re: %S A1 33,52,1716,...

[Hans Havermann]

On Jan 7, 2009, at 10:00 PM, zak seidov wrote:

> 1. is the sequence finite?
> 2. what about other cases of "a" in "is equal to n+a" with more terms?
> 2a. (note that in case a=1 there are only 3 terms cf. A153874)

> %S A000001 33,52,1716,14352,32352,1166352
> %N A000001 Numbers n such that square of product of n's digits is  
> equal to n+48.


I assume you have run through a few cases of "a" to come up with your  
six-solution a=48. I don't know how to prove these sequences finite,  
but a quick run through -100 < a < 100 shows a five-solution a=74 and  
four-solutions for a = -38, -25, -17, -15, 32, 59*, 68*, 72*, 75. (One  
of the solutions in each of the three * cases is negative.)

While large solutions appear to be rare, a glance at the following  
suggests that there are "regions" where solutions are possible:

a = -95   {2264832171464196191}
a = -37   {361, 2264832171464196133}
a = -23   {2264832171464196119}

Might there be a larger region?