A084405 Primes whose sum of factorials of digits is also prime.

2, 11, 13, 31, 101, 163, 313, 331, 431, 503, 613, 631, 1021, 1201, 1223, 1433, 1439, 1453, 1483, 1493, 1543, 1567, 1657, 1663, 1667, 1669, 1753, 1777, 1789, 1879, 1987, 1999, 2011, 2111, 2203, 2213, 2221, 3049, 3163, 3221, 3313, 3331, 3361, 3413, 3461, 3491

Offset: 1

Author

Jason Earls, Jun 24 2003

Keywords

• base
• easy
• nonn

Examples

			a(10)=503, a prime, and 5! + 0! + 3! = 127, a prime.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A061602.

Programs

Mathematica

Select[Prime[Range[500]],PrimeQ[Total[IntegerDigits[#]!]]&] (* Harvey P. Dale, Mar 20 2016 *)

PARI

{digitsumfac(n)=local(s, d); s=0; while(n>0,d=divrem(n,10); n=d[1]; s=s+d[2]!); s}
{facp(m)=local(ct,sr); ct=0; sr=0; forprime(p=2,m, if(isprime(digitsumfac(p)),ct++; print1(p," "); sr+=(1.0/p); )); print(); print("Found: "ct" primes < "m); print("Sum of reciprocals = "sr); }

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 isprime(n) and isprime(f(n))
print([k for k in range(3500) if ok(k)]) # Michael S. Branicky, Feb 11 2023