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

%I A173689 #19 Feb 23 2023 11:53:14
%S A173689 0,1,2,3,4,5,6,7,8,9,122,202,212,220,221,244,424,442,1000,1001,1010,
%T A173689 1011,1100,1101,1110,1111,2222,3333,3444,4344,4434,4443,4444,5555,
%U A173689 6666,6677,6767,6776,6888,7667,7676,7766,7777,8688,8868,8886,8888,9999
%N A173689 Numbers m such that the sum of square of factorial of decimal digits is square.
%C A173689 Let the decimal expansion of m = d(0)d(1)...d(p). Numbers such that Sum_{k=0..p} (d(k)!)^2 is square.
%H A173689 Jinyuan Wang, <a href="/A173689/b173689.txt">Table of n, a(n) for n = 1..1487</a>
%e A173689 a(16) = 244 is in the sequence because (2!)^2 + (4!)^2 + (4!)^2 = 1156 = 34^2.
%p A173689 with(numtheory):for n from 0 to 10000 do:l:=length(n):n0:=n:s:=0:for m from
%p A173689   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
%p A173689   floor(q)= q then printf(`%d, `,n):else fi:od:
%t A173689 Select[Range[0,10000],IntegerQ[Sqrt[Total[(IntegerDigits[#]!)^2]]]&] (* _Harvey P. Dale_, Dec 19 2011 *)
%o A173689 (Python)
%o A173689 from itertools import count, islice, combinations_with_replacement
%o A173689 from math import factorial
%o A173689 from sympy.ntheory.primetest import is_square
%o A173689 from sympy.utilities.iterables import multiset_permutations
%o A173689 def A173689_gen(): # generator of terms
%o A173689     yield 0
%o A173689     for l in count(0):
%o A173689         for i in range(1,10):
%o A173689             fi = factorial(i)**2
%o A173689             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))))
%o A173689 A173689_list = list(islice(A173689_gen(),50)) # _Chai Wah Wu_, Feb 23 2023
%Y A173689 Cf. A173687, A173688.
%K A173689 nonn,base
%O A173689 1,3
%A A173689 _Michel Lagneau_, Nov 25 2010
%E A173689 Offset changed to 1 by _Jinyuan Wang_, Feb 26 2020