

A343828


Numbers which are the product of two Sprimes (A057948) in exactly three ways.


4



4389, 5313, 7161, 9177, 9933, 10857, 12369, 13629, 14421, 14973, 15477, 16401, 17157, 18249, 18753, 19173, 19437, 20769, 22701, 23529, 23541, 23793, 24717, 26733, 26961, 27993, 28329, 28497, 29337, 29469, 30261, 30597, 31521, 32109, 32361, 32637, 33117, 33649
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

There exist numbers which are the product of two Sprimes in exactly 1, 2, and 3 ways.
An Sprime is either a prime of the form 4k+1 or a semiprime of the form (4k+3)*(4m+3). That means the maximum number of prime factors that a number factorizable into two Sprimes can have is four (all 4k + 3), and those can be combined into Sprimes in at most three distinct ways.  Gleb Ivanov, Dec 07 2021


LINKS



FORMULA



EXAMPLE

9177 = 21*437 = 57*161 = 69*133 which are all Sprimes (A057948), and admits no other SPrime factorizations.
4389 = (3*7)*(11*19) = (3*11)*(7*19) = (3*19)*(7*11); 3,7,11,19 are the smallest primes of the form 4k + 3.


PROG

isok(n) = sumdiv(n, d, (d<=n/d) && is(d) && is(n/d)) == 3; \\ Michel Marcus, May 01 2021


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



