A061602 Sum of factorials of the digits of n.

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 2, 2, 3, 7, 25, 121, 721, 5041, 40321, 362881, 3, 3, 4, 8, 26, 122, 722, 5042, 40322, 362882, 7, 7, 8, 12, 30, 126, 726, 5046, 40326, 362886, 25, 25, 26, 30, 48, 144, 744, 5064, 40344, 362904, 121, 121, 122, 126

Offset: 0

Author

Amarnath Murthy, May 19 2001

Keywords

• nonn
• base
• easy

Comments

Numbers n such that a(n) = n are known as factorions. It is known that there are exactly four of these [in base 10]: 1, 2, 145, 40585. - Amarnath Murthy

The sum of factorials of the digits is the same for 0, 1, 2 in any base. - Alonso del Arte, Oct 21 2012

Examples

			a(24) = (2!) + (4!) = 2 + 24 = 26.
a(153) = 127 because 1! + 5! + 3! = 1 + 120 + 6 = 127.
		

Links

• Harry J. Smith and Indranil Ghosh, Table of n, a(n) for n = 0..10000 (first 1001 terms from Harry J. Smith)
• Project Euler, Problem 74: Digit factorial chains
• H. J. J. te Riele, Iteration of number-theoretic functions, Nieuw Archief v. Wiskunde, (4) 1 (1983), 345-360. See Example I.1.b.
• Eric Weisstein's World of Mathematics, Factorion.

Crossrefs

Cf. A061603, A108911, A193163, A165451 (places of primes).

See also A014080, A214285, A254499, A306955.

Programs

Magma

a061602:=func< n | n eq 0 select 1 else &+[ Factorial(d): d in Intseq(n) ] >; [ a061602(n): n in [0..60] ]; // Klaus Brockhaus, Nov 23 2010

Maple

A061602 := proc(n)
        add(factorial(d),d=convert(n,base,10)) ;
end proc: # R. J. Mathar, Dec 18 2011

Mathematica

a[n_] := Total[IntegerDigits[n]! ]; Table[a[n], {n, 1, 53}] (* Saif Hakim (saif7463(AT)gmail.com), Apr 23 2006 *)

PARI

a(n) = { if(n==0, 1, my(d=digits(n)); sum(i=1, #d, d[i]!)) } \\ Harry J. Smith, Jul 25 2009

Python

import math
def A061602(n):
    s=0
    for i in str(n):
        s+=math.factorial(int(i))
    return s # Indranil Ghosh, Jan 11 2017

R

i=0
values <- c()
while (i<1000) {
  values[i+1] <- A061602(i)
  i=i+1
}
plot(values)
A061602 <- function(n) {
  sum=0;
  numberstring <- paste0(i)
  numberstring_split <- strsplit(numberstring, "")[[1]]
  for (number in numberstring_split) {
    sum = sum+factorial(as.numeric(number))
  }
  return(sum)
}
# Raphaël Deknop, Nov 08 2021

Extensions

Corrected and extended by Vladeta Jovovic, May 19 2001

Link and amended comment by Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 12 2004