A173687 Numbers m such that the sum of factorial of the decimal digits of m is square.

0, 1, 14, 15, 17, 22, 40, 41, 45, 50, 51, 54, 70, 71, 102, 112, 120, 121, 123, 132, 144, 156, 165, 200, 201, 203, 210, 211, 213, 230, 231, 302, 312, 320, 321, 334, 343, 404, 414, 433, 440, 441, 457, 475, 506, 516, 547, 560, 561, 574, 605, 615, 650, 651, 745, 754, 1000

Offset

1,3

Comments

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

Same as A130687 except for the additional 0. - Georg Fischer, Oct 12 2018

Links

Jinyuan Wang, Table of n, a(n) for n = 1..9700

Example

17 is in the sequence because 1! + 7! = 1 + 5040 = 71^2.

Maple

with(numtheory):for n from 0 to 1000 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!:od: q:=sqrt(s):if
  floor(q)= q then printf(`%d, `, n):else fi:od:~

Mathematica

Select[Range[0, 1000], IntegerQ[Sqrt[Total[IntegerDigits[#]!]]]&] (* Jinyuan Wang, Feb 26 2020 *)

Prog

(Magma) [ n: n in [0..1000] | n eq 0 or IsSquare(&+[ Factorial(d): d in Intseq(n) ]) ];
(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 A173687_gen(): # generator of terms
    yield 0
    for l in count(0):
        for i in range(1, 10):
            fi = factorial(i)
            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(factorial, j))))
A173687_list = list(islice(A173687_gen(), 50)) # Chai Wah Wu, Feb 23 2023

Crossrefs

Sequence in context: A293792 A308974 A130687 * A114962 A278979 A220671

Adjacent sequences: A173684 A173685 A173686 * A173688 A173689 A173690

Keyword

nonn,base

Author

Michel Lagneau, Nov 25 2010

Status

approved