A254499 -id:A254499 - 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.

A306955 Let f map k to the sum of the factorials of the digits of k (A061602); sequence lists numbers such that f(f(f(k)))=k.

Original entry on oeis.org

1, 2, 145, 169, 1454, 40585, 363601

Offset: 1

N. J. A. Sloane, Mar 17 2019

Kiss showed that there are no further terms and in fact there are no further cycles other than those shown in A014080 and A254499.

			The map f sends 169 to 363601 to 1454 to 169 ...

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.
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.

Cf. A061602.

The fixed points and loops of length 2 can be found in A014080, A214285, and A254499.

f[k_] := Total[IntegerDigits[k]!]; Select[Range[400000], Nest[f, #, 3] == # &] (* Amiram Eldar, Mar 17 2019 *)

a061602(n) = my(d=digits(n)); sum(i=1, #d, d[i]!)
is(n) = a061602(a061602(a061602(n)))==n \\ Felix Fröhlich, May 18 2019

A343269 a(n) is the smallest integer whose orbit length is n under iteration of the map r -> A061602(r).

Original entry on oeis.org

1, 0, 169, 78, 69, 26, 24, 4, 22, 5, 122, 25, 14, 127, 6, 3, 12, 33, 136, 256, 57, 247, 148, 38, 1478, 368, 79, 1458, 48, 44, 29, 7, 13, 34, 9, 8, 23, 234, 37, 337, 58, 46, 139, 138, 369, 239, 267, 36, 334, 289, 3555, 49, 144, 45, 229, 2569, 22888, 136789, 334479, 1479, 1233466

Offset: 1

Lamine Ngom, Apr 10 2021

A303935 provides the orbit's lengths, i.e., the number of needed steps, starting from a given number, to reach a value that already exists in trajectory.
This sequence is infinite. Actually, given a number x whose orbit's length is k, one can always build a number y whose orbit's length is (k+1).
For instance, just consider either the number 10^(x-1), or Rx (the repunit of length x), or any other x-digit binary string, all of them leading to the number x after application of the mapping function: A061602(y) = x.
Indeed, none of them will correspond to the smallest integer m such that A303935(m) = k + 1.
In fact, it becomes computationally hard to determine further terms since, as in the Collatz mapping function and other similar problems, there is no predictable way to define the exact complete path without calculating all intermediary orbit's components until one reaches a previously calculated or encountered number.
a(59) = 334479, a(60) = 1479, a(61) = 1233466, next terms = ?

			a(6) = 26 because A303935(26) = 6, and 26 is the smallest nonnegative integer m such that A303935(m) = 6.

Cf. A303935 (orbit's length), A061602 (sum of factorials of digits), A014080 (factorions).

Cf. A193163, A214285, A254499, A188283, A244090.

cp's OEIS Frontend

A061602 Sum of factorials of the digits of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

R

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

J

Mathematica

Python

Formula

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A306955 Let f map k to the sum of the factorials of the digits of k (A061602); sequence lists numbers such that f(f(f(k)))=k.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A343269 a(n) is the smallest integer whose orbit length is n under iteration of the map r -> A061602(r).

Views

Author

Keywords

Comments

Examples

Crossrefs