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

1,1

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.

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A165451, A173687, A173689.

Sequence in context: A297266 A296707 A297140 * A008709 A008708 A102490

Adjacent sequences: A173685 A173686 A173687 * A173689 A173690 A173691

Keyword

nonn,base

Author

Michel Lagneau, Nov 25 2010

Status

approved