A014597 Numbers k such that k^2 is a sum of distinct factorials.

1, 3, 5, 11, 12, 27, 29, 71, 72, 213, 215, 603, 635, 1917, 1183893

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• more
• hard
• nice

Comments

a(16)^2 > 48! (about 1.24139*10^61), if it exists. - Jon E. Schoenfield, Aug 04 2006

A197183(a(n)) = 1. - Reinhard Zumkeller, Dec 04 2011

a(16) > 4.3*10^55 if it exists. - Bert Dobbelaere, Sep 16 2020

Examples

			1183893^2 = 1! + 2! + 3! + 7! + 8! + 9! + 10! + 11! + 12! + 13! + 14! + 15!.
2 is not a member since 4 is not a sum of distinct factorials.
		

References

• Posting by Dan Hoey to math-fun mailing list.

Links

• Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
• Eric Weisstein's World of Mathematics, Factorial

Crossrefs

Cf. A025494, A051761, A059589.

Programs

Haskell

import Data.List (elemIndices)
a014597 n = a014597_list !! (n-1)
a014597_list = tail $ elemIndices 1 $ map a197183 [0..]
-- Reinhard Zumkeller, Dec 04 2011

Mathematica

ok[n_] := (k=1; ff={}; While[k! < n^2, AppendTo[ff, k!]; k++]; xx = Array[x, Length[ff]]; Reduce[And @@ (0 <= # <= 1 & /@ xx) && n^2 == xx.ff, xx, Integers] =!= False); ok[1] = True; Reap[Do[If[ok[n], Print[n]; Sow[n]], {n, 1, 2*10^6}]][[2, 1]] (* Jean-François Alcover, Jul 16 2012 *)

Python

from math import factorial, isqrt
from itertools import chain, combinations
from sympy.ntheory.primetest import is_square
fac =[factorial(n) for n in range(1, 16)] # raise 16 to search higher
def powerset(s): # skipping empty set
  return chain.from_iterable(combinations(s, r) for r in range(1, len(s)+1))
gen = (isqrt(sum(s)) for s in powerset(fac) if is_square(sum(s)))
print(sorted(set(gen))) # Michael S. Branicky, Jan 03 2021

Extensions

15th term from Jud McCranie, who remarks that there are no others involving terms < 21!.