A306955 -id:A306955 - cp's OEIS frontend

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

Amarnath Murthy, May 19 2001

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

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

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.

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

See also A014080, A214285, A254499, A306955.

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

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

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

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

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

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

Corrected and extended by Vladeta Jovovic, May 19 2001

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

A014080 Factorions: equal to the sum of the factorials of their digits in base 10 (cf. A061602).

Original entry on oeis.org

1, 2, 145, 40585

Offset: 1

David W. Wilson

Poole (1971) showed that there are no further terms. - N. J. A. Sloane, Mar 17 2019
Base 6 also has four factorions, as does base 15. - Alonso del Arte, Oct 20 2012
This is row 10 of the table A193163. - M. F. Hasler, Nov 25 2015

			1! + 4! + 5! = 1 + 24 + 120 = 145, so 145 is in the sequence.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 145, p. 50, Ellipses, Paris 2008.
P. Kiss, A generalization of a problem in number theory, Math. Sem. Notes Kobe Univ., 5 (1977), no. 3, 313-317. MR 0472667 (57 #12362).
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see pp. 68, 305.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 38, 62-63.
Joe Roberts, "The Lure of the Integers", page 35.
D. Wells, Curious and interesting numbers, Penguin Books, p. 125.

Project Euler, Problem 34: Digit factorials
P. Kiss, A generalization of a problem in number theory, [Hungarian], Mat. Lapok, 25 (No. 1-2, 1974), 145-149.
G. D. Poole, Integers and the sum of the factorials of their digits, Math. Mag., 44 (1971), 278-279, [JSTOR].
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

Cf. A061602, A193163, A214285, A254499, A306955.

(#~ (= +/@:!@:("."0)@":"0)) i.1e5 NB. Stephen Makdisi, May 14 2016

Select[Range[50000], Plus @@ (IntegerDigits[ # ]!) == # &] (* Alonso del Arte, Jan 14 2008 *)

from itertools import count, islice
def A014080_gen(): # generator of terms
    return (n for n in count(1) if sum((1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880)[int(d)] for d in str(n)) == n)
A014080_list = list(islice(A014080_gen(),4)) # Chai Wah Wu, Feb 18 2022

If n has digits (d1,d2,...,dk) base 10, then n is on this list if and only if n = d1! + d2! + ... + dk!.

A214285 List of amicable sums-of-factorial-of-digits pairs (A,B): A equals the sum of the factorials of B's digits in base 10, and vice versa.

Original entry on oeis.org

871, 45361, 872, 45362

Offset: 1

Jaeyool Park, Jul 10 2012

Number pairs (A,B), A <> B, such that A061602(A)=B and A061602(B)=A, indicating where the mapping of A to the sum of the factorials of its digits has a cycle of length 2.

Peter Kiss (1977) showed there are no further terms. - N. J. A. Sloane, Mar 17 2019

			8! + 7! + 1! = 45361, 4! + 5! + 3! + 6! + 1! = 871.

P. Kiss, A generalization of a problem in number theory, Math. Sem. Notes Kobe Univ., 5 (1977), no. 3, 313-317. MR 0472667 (57 #12362).

P. Kiss, A generalization of a problem in number theory, [Hungarian], Mat. Lapok, 25 (No. 1-2, 1974), 145-149.
Jaeyool Park, Blog [in Korean].
G. D. Poole, Integers and the sum of the factorials of their digits, Math. Mag., 44 (1971), 278-279, [JSTOR].
Project Euler, Problem 74-Digit factorial chains
Shoei Takahashi, Unchone Lee, Hikaru Manabe, Aoi Murakami, Daisuke Minematsu, Kou Omori, and Ryohei Miyadera, Curious Properties of Iterative Sequences, arXiv:2308.06691 [math.GM], 2023.
H. J. J. te Riele, Iteration of number-theoretic functions, Nieuw Archief v. Wiskunde, (4) 1 (1983), 345-360. See Example I.1.b.

Cf. A061602, A014080, A188284, A254499, A306955.

A254499 Amicable factorions.

Original entry on oeis.org

1, 2, 145, 871, 872, 40585, 45361, 45362

Offset: 1

Michel Lagneau, Jan 31 2015

The members of a pair of numbers p and q are called amicable factorions if each is equal to the sum of the factorials of the base-10 digits of the other. The only six pairs (p,q) are (1, 1), (2, 2), (145, 145), (871,45361), (872, 45362), (40585, 40585).

Peter Kiss (1977) showed there are no further terms. - N. J. A. Sloane, Mar 17 2019

			871 and 45361 are in the sequence because:
871 => 8!+7!+1! = 40320 +5040 + 1 = 45361;
45361 => 4!+5!+3!+6!+1! = 24 + 120 + 6 + 720 + 1 = 871.

P. Kiss, A generalization of a problem in number theory, Math. Sem. Notes Kobe Univ., 5 (1977), no. 3, 313-317. MR 0472667 (57 #12362).

S. S. Gupta, Sum of the factorials of the digits of integers, Math. Gaz. 88 (512) (2004) 258-261
P. Kiss, A generalization of a problem in number theory, [Hungarian], Mat. Lapok, 25 (No. 1-2, 1974), 145-149.
G. D. Poole, Integers and the sum of the factorials of their digits, Math. Mag., 44 (1971), 278-279, [JSTOR].
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

Cf. A061602, A306955.

A014080 and A214285 are subsets.

Select[Range[10^6], Plus @@ (IntegerDigits[Plus @@ (IntegerDigits[ # ]!) ]!) == # &]

n such that f(f(n))=n, where f(k)=A061602(k).

A308259 a(n) is equal to the sum of the factorials of the digits of a(n-1), initial term is 3.

Original entry on oeis.org

3, 6, 720, 5043, 151, 122, 5, 120, 4, 24, 26, 722, 5044, 169, 363601, 1454, 169, 363601, 1454, 169, 363601, 1454, 169, 363601, 1454, 169, 363601, 1454, 169, 363601, 1454, 169, 363601, 1454, 169, 363601, 1454, 169, 363601, 1454, 169, 363601, 1454, 169, 363601

Offset: 1

Romain Pinot, May 17 2019

From a(14) = 169, the sequence is stuck in the loop 363601, 1454, 169, 363601, 1454, ...

			a(5) = 151, so a(6) = 1! + 5! + 1! = 1 + 120 + 1 = 122.

Shoei Takahashi, Unchone Lee, Hikaru Manabe, Aoi Murakami, Daisuke Minematsu, Kou Omori, and Ryohei Miyadera, Curious Properties of Iterative Sequences, arXiv:2308.06691 [math.GM], 2023.

Cf. A061602, A306955.

NestList[Total@ Map[Factorial, IntegerDigits[#]] &, 3, 36] (* Michael De Vlieger, Aug 21 2023 *)

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A061602 Sum of factorials of the digits of n.

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A014080 Factorions: equal to the sum of the factorials of their digits in base 10 (cf. A061602).

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A214285 List of amicable sums-of-factorial-of-digits pairs (A,B): A equals the sum of the factorials of B's digits in base 10, and vice versa.

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A254499 Amicable factorions.

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A308259 a(n) is equal to the sum of the factorials of the digits of a(n-1), initial term is 3.

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