A173688 Numbers m such that the sum of square of factorial of decimal digits is prime.

10, 11, 12, 13, 14, 15, 19, 20, 21, 30, 31, 40, 41, 50, 51, 90, 91, 100, 101, 110, 111, 123, 132, 133, 134, 135, 138, 143, 144, 147, 153, 156, 158, 165, 168, 169, 174, 177, 183, 185, 186, 196, 203, 213, 230, 231, 302, 303, 304, 305, 308, 312, 313, 314, 315, 318, 320

Offset: 1

Author

Michel Lagneau, Nov 25 2010

Keywords

• nonn
• base

Comments

Let the decimal expansion of m = d(0)d(1)...d(p). Numbers such that Sum_{k=0..p} (d(k)!)^2 is prime.

Examples

			a(5) = 14 is in the sequence because (1!)^2 + (4!)^2 = 1 + 24^2 = 577 and 577 is prime.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..6821

Crossrefs

Cf. A165451, A173687, A173689.

Programs

Maple

with(numtheory):for n from 1 to 500 do:l:=length(n):n0:=n:s:=0:for m from 1
  to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s:=s+(u!)^2:od: if type(s,prime)=true
  then printf(`%d, `,n):else fi:od:

Mathematica

Select[Range[400],PrimeQ[Total[(IntegerDigits[#]!)^2]]&]  (* Harvey P. Dale, Mar 23 2011 *)

Python

from itertools import count, islice, combinations_with_replacement
from math import factorial
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A173688_gen(): # generator of terms
    for l in count(0):
        for i in range(1,10):
            fi = factorial(i)**2
            yield from sorted(int(str(i)+''.join(map(str,k))) for j in combinations_with_replacement(range(10), l) for k in multiset_permutations(j) if isprime(fi+sum(map(lambda n:factorial(n)**2,j))))
A173688_list = list(islice(A173688_gen(),50)) # Chai Wah Wu, Feb 23 2023