A014080 -id:A014080 - cp's OEIS frontend

A188283 Numbers k such that iterations for the map r -> A061602(r) starting with k ends with a fixed point (factorions A014080).

0, 1, 2, 10, 11, 145, 154, 223, 232, 322, 405, 415, 450, 451, 504, 514, 540, 541, 569, 596, 659, 695, 956, 965, 1023, 1032, 1123, 1132, 1203, 1213, 1223, 1230, 1231, 1232, 1302, 1312, 1320, 1321, 1322, 1449, 1494, 1569, 1596, 1659, 1695, 1944, 1956, 1965, 2003

Offset: 1

Jaroslav Krizek, Mar 26 2011

If k is a term, then (10^k - 1)/9 is also a term. - Jinyuan Wang, Nov 07 2020

			Number 405 is in sequence because 405 -> 145 -> 145 -> ...

Supersequence of A014080 (factorions).

Cf. A061602, A173447, A188284.

is(k) = {my(t=k, v=List([k])); while(t=sum(i=1, #d=digits(t), d[i]!), if(t==v[#v], return(1), if(sum(i=1, #v-1, t==v[i]), return(0))); listput(v, t)); } \\ Jinyuan Wang, Nov 07 2020

Missing terms a(1) and a(8)-a(10) added by Jaroslav Krizek, Jan 28 2012

Name corrected and more terms from Jinyuan Wang, Nov 07 2020

A244090 Numbers n such that n is a factorion (A014080, equal to the sum of the factorials of its digits), in at least one base b.

Original entry on oeis.org

1, 2, 7, 25, 26, 48, 49, 121, 122, 144, 145, 240, 721, 722, 726, 1440, 1441, 1442, 5041, 5042, 5162, 5760

Offset: 1

Anthony Sand, Jun 20 2014

The bases in which n = func(n) are 2, 2, 4, 6, 6, 11, 5, 24, 24, 28, 10, 47, 120, 120, 240, 239, 15, 15, 720, 720, 27, 822. Note multiple bases for some n, e.g. 25 = 4! + 1! in base 6 and 25 = 1! + 4! in base 21; 721 = 6! + 1! in base 120 and 721 = 1! + 6! in base 715.

			1 = 1! = 1 (base>=2).
2 = 1! + 0! = 1 + 1 = 10 (b=2).
7 = 1! + 3! = 1 + 6 = 13 (b=4).
25 = 4! + 1! = 24 + 1 = 41 (b=6).
26 = 4! + 2! = 24 + 2 = 42 (b=6).
48 = 4! + 4! = 24 + 24 = 44 (b=11).
49 = 1! + 4! + 4! = 1 + 24 + 24 = 144 (b=5).
121 = 5! + 1! = 120 + 1 = 51 (b=24).
122 = 5! + 2! = 120 + 2 = 52 (b=24).
144 = 5! + 4! = 120 + 24 = 54 (b=28).
145 = 1! + 4! + 5! = 1 + 24 + 120 (b=10).
240 = 5! + 5! = 120 + 120 = 55 (b=47).

Wikipedia, Factorion

Cf. A193163, A014080.

isok(n) = {if (n==1, return (1)); for (b=2, n, d = digits(n, b); if (sum(i=1, #d, d[i]!) == n, return (1));); return (0);} \\ Michel Marcus, Jun 21 2014

s = 0; for digit(i=1..j) of n in base b, s = s + digit(i)!.

A061602 Sum of factorials of the digits of n.

Original entry on oeis.org

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

A193163 Irregular table read by rows, in which row n lists the factorions in base n, for n >= 2.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 7, 1, 2, 49, 1, 2, 25, 26, 1, 2, 1, 2, 1, 2, 41282, 1, 2, 145, 40585, 1, 2, 26, 48, 40472, 1, 2, 1, 2, 519326767, 1, 2, 12973363226, 1, 2, 1441, 1442, 1, 2, 2615428934649, 1, 2, 40465, 43153254185213, 43153254226251, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 2

Karl W. Heuer, Aug 06 2011

Because 1 and 2 are factorions in any base, these mark the start of a new row.

			49 is in row 5 because 49 = 1! + 4! + 4! and is "144" in base 5.
The first few rows are:
1, 2 (binary)
1, 2 (ternary)
1, 2, 7 (quartal)
1, 2, 49 (quintal)
1, 2, 25, 26 (hexal)
1, 2 (heptal)
1, 2 (octal)
1, 2, 41282 (nonal)
1, 2, 145, 40585 (decimal)
1, 2, 26, 48, 40472 (undecimal)
1, 2 (duodecimal)
1, 2, 519326767, etc.

Karl W. Heuer, Rows n = 2..39, flattened (each row starts with 1 and 2)
Eric Weisstein's World of Mathematics, Factorion

Cf. A014080, A166623.

A108911 Difference between n and the sum of the factorials of its digits.

Original entry on oeis.org

0, 0, -3, -20, -115, -714, -5033, -40312, -362871, 8, 9, 9, 6, -11, -106, -705, -5024, -40303, -362862, 17, 18, 18, 15, -2, -97, -696, -5015, -40294, -362853, 23, 24, 24, 21, 4, -91, -690, -5009, -40288, -362847, 15, 16, 16, 13, -4, -99, -698, -5017, -40296, -362855, -71, -70, -70, -73

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 18 2005

Null values are at n = 1, 2, 145, 40585 (A014080). Twin values are at n = 1, 2; 11, 12; 21, 22; ... 10*i + 1, 10*i + 2. Not in sequence: 7, 10, 14, ... Nice polar diagrams repeating themselves with normalized angle to 9! and radius = a(n).

The sequence can be seen as the difference between the natural numbers in the decimal system (n_dec = N0*(10^0) + N1*(10^1) + N2*(10^2)...) and their values in a non-positional number system based on the factorials of the digits (n_fact = N0*(N0 - 1)! + N1*(N1 - 1)! + N2*(N2 - 1)! ...). See also A111095. Note that a(np) - a(n) is congruent to 0 mod 9 if n and np are different for the permutation of the digits. Example (a(5971) - a(1957))/9 = 446. The property can be easily derived by remembering that np - n is congruent to 0 mod 9. - Giorgio Balzarotti, Oct 15 2005

			For n = 35, a(35) = -91 because 35 - (3! + 5!) = 35 - (6 + 120) = -91.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005096, A014080, A061602.

[n-&+[Factorial(d): d in Intseq(n)]: n in [1..60]]; // Bruno Berselli, Oct 25 2018

a:= n-> n-add(i!, i=convert(n, base, 10)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 24 2018

f[n_] := n - Plus @@ Factorial /@ IntegerDigits[n]; Table[f[n], {n, 53}] (* Ray Chandler, Jul 24 2005 *)

a(n) = my(d = digits(n)); n - sum(i=1, #d, d[i]!); \\ Michel Marcus, Apr 21 2014

a(n) = n - (N0! + N1! + N2! + ...) if n = N0*10^0 + N1*10^1 + N2*10^2 ...

a(n) = n - A061602(n). - Michel Marcus, Apr 21 2014

Extended by Ray Chandler, Jul 24 2005

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).

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

A173573 3-Factorions: equal to the sum of the triple factorials of their digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 81, 82, 83, 84

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Feb 22 2010

Sequence is complete. - Giovanni Resta, Mar 21 2013

			81 -> 8!!! + 1!!! = 8*5*2 + 1 = 80 + 1 = 81.

Cf. A007661, A014080, A097653, A173574-A173577

P:=proc(n,m) local a,b,i,j,k,x,w; for i from 1 by 1 to n do a:=0; b:=0; w:=0; k:=i; while k>0 do w:=k-(trunc(k/10)*10); j:=w; x:=w-m; if w=0 then b:=1; else while x>0 do j:=j*x; x:=x-m; od; b:=j; fi; a:=a+b; k:=trunc(k/10); od; if a=i then lprint(i,a); fi; od; end: P(1000,3);

stfQ[n_]:=n==Total[Times@@Range[#,1,-3]&/@IntegerDigits[n]]; Select[Range[100],stfQ] (* Harvey P. Dale, May 31 2023 *)

A173577 7-Factorions: equal to the sum of the 7-fold factorials of their digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 19

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Feb 22 2010

			19 -> 1!!!!!!! + 9!!!!!!! = 1 + 9*2 = 1 + 18 = 19.

A014080, A097653, A173573-A173576

P:=proc(n,m) local a,b,i,j,k,x,w; for i from 1 by 1 to n do a:=0; b:=0; w:=0; k:=i; while k>0 do w:=k-(trunc(k/10)*10); j:=w; x:=w-m; if w=0 then b:=1; else while x>0 do j:=j*x; x:=x-m; od; b:=j; fi; a:=a+b; k:=trunc(k/10); od; if a=i then lprint(i,a); fi; od; end: P(1000,7);

Definition corrected by Paolo P. Lava, Feb 24 2010

cp's OEIS Frontend

A188283 Numbers k such that iterations for the map r -> A061602(r) starting with k ends with a fixed point (factorions A014080).

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A244090 Numbers n such that n is a factorion (A014080, equal to the sum of the factorials of its digits), in at least one base b.

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A061602 Sum of factorials of the digits of n.

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A193163 Irregular table read by rows, in which row n lists the factorions in base n, for n >= 2.

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A108911 Difference between n and the sum of the factorials of its digits.

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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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