A130687 -id:A130687 - cp's OEIS frontend

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

Michel Lagneau, Nov 25 2010

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

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

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

[ n: n in [0..1000] | n eq 0 or IsSquare(&+[ Factorial(d): d in Intseq(n) ]) ];

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:~

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

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

A130688 Numbers n with following property: suppose n^6 = d1 d2 d3 ...dk in decimal; then d1! + d2! + ... + dk! is a square.

Original entry on oeis.org

1, 6, 747, 2802, 10000, 10256, 11876, 13875, 14623, 14710, 17117, 18090, 23919, 26569, 34282, 35402, 40515, 41202, 41850, 42195, 44684, 48396, 54698, 58509, 59293, 59644, 59900, 65502, 67795, 74004, 75320, 79593, 82677, 82713, 83402

Offset: 1

Yalcin Aktar, Jun 30 2007

Numbers n such that n^6 is in A130687.

			a(2) = 6, because 6^6 = 46656, and (4!+6!+6!+5!+6!)^(1/2) = 48 is an integer.

Cf. A130687, A089185, A126076, A126077, A126078.

A061602 := proc(n) local digs ; digs := convert(n,base,10) ; add(factorial(op(i,digs)),i=1..nops(digs)) ; end: isA130687 := proc(n) RETURN(issqr(A061602(n))) ; end: isA130688 := proc(n) RETURN(isA130687(n^6)) ; end: for n from 1 to 130000 do if isA130688(n) then printf("%d, ",n) ; fi : od:

for(n=1,10^5,m=n^6;s=0;while(m,s+=(m%10)!;m\=10);if(issquare(s),print1(n",")))

Edited by R. J. Mathar and Martin Fuller, Jul 13 2007

cp's OEIS Frontend

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

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