MathDB

Find all natural numbers

n

for which the following

5

conditions hold:

(1)

n

is not divisible by any perfect square bigger than

1

(2)

n

has exactly one prime divisor of the form

4k+3

k\in \mathbb{N}_0

(3)

Denote by

S(n)

the sum of digits of

n

and

d(n)

as the number of positive divisors of

n

. Then we have that

S(n)+2=d(n)

(4)

n+3

is a perfect square.

(5)

n

does not have a prime divisor which has

4

or more digits.

Problem Statement