A358998 Nonprimes whose sum of factorials of digits is a prime.

10, 12, 20, 21, 30, 100, 110, 111, 122, 133, 134, 135, 136, 143, 153, 155, 178, 187, 202, 212, 220, 221, 303, 304, 305, 306, 314, 315, 316, 330, 340, 341, 350, 351, 360, 361, 403, 413, 430, 505, 513, 515, 530, 531, 550, 551, 603, 630, 708, 718, 780, 781, 807

Offset: 1

Author

Carole Dubois, Feb 11 2023

Keywords

• base
• easy
• nonn

Examples

			134 is in the sequence because it is not prime and 1! + 3! + 4! = 1 + 6 + 24 = 31 which is a prime number.
202 is also in the sequence because it is not prime and 2! + 0! + 2! = 5 prime.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A061602, A084405 (Primes such that the sum of the factorials of the digits is also prime).

Programs

Mathematica

Select[Range[1000], ! PrimeQ[#] && PrimeQ[Total[IntegerDigits[#]!]] &] (* Amiram Eldar, Feb 11 2023 *)

Python

from sympy import isprime
from math import factorial
S=[]
nomb=200
i=0
while len(S) < nomb:
    i+=1
    if isprime(i):
        continue
    som=0
    for j in range(len(str(i))):
        som+=factorial(int(ix[j]))
    if  not isprime(som):
        continue
    S.append(i)

Python

from sympy import isprime
from math import factorial
def f(n): return sum(factorial(int(d)) for d in str(n))
def ok(n): return not isprime(n) and isprime(f(n))
print([k for k in range(900) if ok(k)]) # Michael S. Branicky, Feb 11 2023

Formula

A165451 INTERSECT A002808. - R. J. Mathar, Feb 23 2023