A173687 -id:A173687 - cp's OEIS frontend

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

Michel Lagneau, Nov 25 2010

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

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

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

Cf. A165451, A173687, A173689.

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:

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

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

A173689 Numbers m such that the sum of square of factorial of decimal digits is square.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 122, 202, 212, 220, 221, 244, 424, 442, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 2222, 3333, 3444, 4344, 4434, 4443, 4444, 5555, 6666, 6677, 6767, 6776, 6888, 7667, 7676, 7766, 7777, 8688, 8868, 8886, 8888, 9999

Offset: 1

Michel Lagneau, Nov 25 2010

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

			a(16) = 244 is in the sequence because (2!)^2 + (4!)^2 + (4!)^2 = 1156 = 34^2.

Jinyuan Wang, Table of n, a(n) for n = 1..1487

Cf. A173687, A173688.

with(numtheory):for n from 0 to 10000 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: q:=sqrt(s):if
  floor(q)= q then printf(`%d, `,n):else fi:od:

Select[Range[0,10000],IntegerQ[Sqrt[Total[(IntegerDigits[#]!)^2]]]&] (* Harvey P. Dale, Dec 19 2011 *)

from itertools import count, islice, combinations_with_replacement
from math import factorial
from sympy.ntheory.primetest import is_square
from sympy.utilities.iterables import multiset_permutations
def A173689_gen(): # generator of terms
    yield 0
    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 is_square(fi+sum(map(lambda n:factorial(n)**2,j))))
A173689_list = list(islice(A173689_gen(),50)) # Chai Wah Wu, Feb 23 2023

Offset changed to 1 by Jinyuan Wang, Feb 26 2020

cp's OEIS Frontend

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

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