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

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

1,3

Comments

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

Links

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

Example

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

Maple

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:

Mathematica

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

Prog

(Python)
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

Crossrefs

Cf. A173687, A173688.

Sequence in context: A134853 A193407 A134810 * A004871 A059405 A191872

Adjacent sequences: A173686 A173687 A173688 * A173690 A173691 A173692

Keyword

nonn,base

Author

Michel Lagneau, Nov 25 2010

Extensions

Offset changed to 1 by Jinyuan Wang, Feb 26 2020

Status

approved