A014080 -id:A014080 - cp's OEIS frontend

A139180 Smallest number m greater than 1 such that m=(Sum_{i=1..k} d_i!)^n where d_1 d_2 ... d_k is the decimal expansion of m, or 0 if no m exists.

2, 172603687936, 3910825640696988096, 396867211295273131775761, 13586030059671762261567625351, 4218282992523473247642107344280879104, 5572272365289149102816175194628254483025277

Offset: 1

Farideh Firoozbakht, Apr 22 2008

A139181(n)=A139180(n)^(1/n).

			172603687936=(1!+7!+2!+6!+0!+3!+6!+8!+7!+9!+3!+6!)^2

Cf. A014080, A137604, A137605.

A173574 4-Factorions: equal to the sum of the quadruple factorials of their digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 49

Offset: 1

Paolo P. Lava, Feb 22 2010

			49 -> 4!!!! + 9!!!! = 4 + 9*5 = 4 +45 = 49.

Cf. A007662, A014080, A097653, A173573, A173575-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(1000,4);

qfd[n_]:=Times@@Range[n,1,-4]; Select[Range[50],Total[qfd/@ IntegerDigits[ #]] == #&] (* Harvey P. Dale, Dec 15 2018 *)

A173575 5-Factorions: equal to the sum of the quintuple factorials of their digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 39

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Feb 22 2010

			39 -> 3!!!!! + 9!!!!! = 3 + 9*4 = 3 + 36 = 39.

A014080, A097653, A173573, A173574, A173576, 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(1000,5);

A173576 6-Factorions: equal to the sum of the 6-ple factorials of their digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 29

Offset: 1

Paolo P. Lava, Feb 22 2010

			29 -> 2!!!!!! + 9!!!!!! = 2 + 9*3 = 2 + 27 = 29.

Cf. A014080, A097653, A173573-A173575, 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(1000,6);

A188284 Finite sequence of numbers n such that iterations for the map r -> A061602(r) starting with n ends with the same number n.

Original entry on oeis.org

1, 2, 145, 169, 871, 872, 1454, 40585, 45361, 45362, 363601

Offset: 1

Jaroslav Krizek, Mar 26 2011

See A173447 = the number of iterations for the map r -> A061602(r), A061602 = sum of factorials of the digits of n.

Superset of A014080 (factorions).

			Number 169 is in sequence because 169 -> 363601 -> 1454 -> 169.

Cf. A188283, A014080, A173447, A061602.

A303935 Size of orbit of n under repeated application of sum of factorial of digits of n.

Original entry on oeis.org

2, 1, 1, 16, 8, 10, 15, 32, 36, 35, 2, 2, 17, 33, 13, 10, 15, 32, 36, 35, 17, 17, 9, 37, 7, 12, 6, 8, 33, 31, 33, 33, 37, 18, 34, 31, 48, 39, 24, 8, 13, 13, 7, 34, 30, 54, 42, 39, 29, 52, 10, 10, 12, 31, 54, 10, 24, 21, 41, 24, 15, 15, 6, 48, 42, 24, 12, 42

Offset: 0

Philippe Guglielmetti, May 03 2018

Numbers n for which a(n)=1 are called factorions (A014080).
Apart from factorions, only 3 cycles exist:
169 -> 363601 -> 1454 -> 169, so a(169) = a(363601) = a(1454) = 3.
871 -> 45361 -> 871, so a(871) = a(45361) = 2.
872 -> 45362 -> 872, so a(872) = a(45362) = 2.
All other n produce a chain reaching either a factorion or a cycle.

			For n = 4, 4!=24, 2!+4!=26, 2!+6!=722, 7!+2!+2!=5044, 5!+0!+4!+4!=169, 1!+6!+9!=363601, 3!+6!+3!+6!+0!+1!=1454, then 1!+4!+5!+4!=169 which already belongs to the chain, so a(4) = length of [4, 24, 26, 722, 5044, 169, 363601, 1454] = 8.

Philippe Guglielmetti, Table of n, a(n) for n = 0..1000
S. S. Gupta, Sum of the factorials of the digits of integers, The Mathematical Gazette, 88-512 (2004), 258-261.
Project Euler, Problem 74: Digit factorial chains

Cf. A061602, A014080 (contains n for which a(n) = 1).

Array[Length@ NestWhileList[Total@ Factorial@ IntegerDigits@ # &, #, UnsameQ, All, 100, -1] &, 68, 0] (* Michael De Vlieger, May 10 2018 *)

f(n) = if (!n, n=1); my(d=digits(n)); sum(k=1, #d~, d[k]!);
a(n) = {my(v = [n], vs = Set(v)); for (k=1, oo, new = f(n); if (vecsearch(vs, new), return (#vs)); v = concat(v, new); vs = Set(v); n = new;);} \\ Michel Marcus, May 18 2018

for n in count(0):
    l=[]
    i=n
    while i not in l:
        l.append(i)
        i=sum(map(factorial,map(int,str(i))))
    print(n,len(l))

A101698 Numbers n such that n = d_1d_1!+d_2d_2!+...+d_k*d_k!, where d_1 d_2 ... d_k is the decimal expansion of n.

Original entry on oeis.org

0, 1, 684817, 3371797

Offset: 1

Farideh Firoozbakht, Dec 18 2004

			3371797 is in the sequence because 3371797 = 3*3!+3*3!+7*7!+1*1!+7*7!+9*9!+7*7!.

Cf. A014080, A097642, A101699.

Do[If[n==Apply[Plus, IntegerDigits[n]*IntegerDigits[n]! ], Print[n]], {n, 26000000}]

A163576 Near-factorions: equal to the sum of the factorials of all but one of their digits in base 10.

Original entry on oeis.org

24, 5761, 5762

Offset: 1

Berend Jan van der Zwaag (b.j.vanderzwaag(AT)utwente.nl), Jul 31 2009

			24 = 4!, 5761 = 7! + 6! + 1!, 5762 = 7! + 6! + 2!

Clifford A. Pickover, "Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning", Oxford University Press, 2003, p. 357.

Cf. A014080.

f = factorial(0:9); n = 0; for m = 1:2177280, d = f(mod(floor(m*10.^(-floor(log10(m)):0)),10)+1); if any(sum(d)-d==m), n = n+1; a(n) = m; end, end

A247227 Numbers that divide the sum of the factorials of their digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 56, 71, 93, 145, 219, 758, 768, 7584, 7684, 9696, 10081, 21993, 40585

Offset: 1

Jaroslav Krizek, Dec 25 2014

Numbers n such that n divides A061602(n).
Finite sequence with 23 terms.
Subsequence of numbers n such that A061602(n) >= n. The largest such number is 1999999.
Sequence of values A061602(a(n)) / a(n): 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 19099, 15, 71, 3902, 1, 1657, 60, 60, 6, 6, 75, 4, 33, 1.
Supersequence of A014080 (whose terms are the numbers n such that A061602(n) / n = 1).

			19 is in the sequence because 19 divides 1! + 9! = 1 + 362880 = 362881; 362881 / 19 = 19099.

Cf. A014080, A061602, A093325, A197181.

[n: n in [1..2000000] | &+[ Factorial(d): d in Intseq(n)] mod n eq 0]

Select[Range[50000], Mod[Plus @@ Factorial[IntegerDigits[#]], #] == 0 &] (* Michael De Vlieger, Dec 26 2014 *)

for(k=1,10^5,d=digits(k);s=sum(i=1,#d,d[i]!);if(!(s%k),print1(k,", "))) \\ Derek Orr, Dec 30 2014

A345405 Integers k such that k = (d1)_c + (d2)_c + ... + (dc)_c, where (d)_c denotes the descending factorial of d, c is the length of k and di is the i-th digit of k in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 86, 15960

Offset: 1

Andrzej Kukla, Jun 18 2021

The descending factorial (d)_c is defined as d*(d-1)*(d-2)*...*(d-c+1).

			(8)_2 + (6)_2 = 8*7 + 6*5 = 56 + 30 = 86, therefore 86 is in the list.

Cf. A014080 (factorions), A068424 (descending factorials), A345406.

q[n_] := Module[{dig = IntegerDigits[n], nd}, nd = Length[dig]; Sum[d!/(d - nd)!, {d, dig}] == n]; Select[Range[0, 16000], q] (* Amiram Eldar, Jun 18 2021 *)

cp's OEIS Frontend

A139180 Smallest number m greater than 1 such that m=(Sum_{i=1..k} d_i!)^n where d_1 d_2 ... d_k is the decimal expansion of m, or 0 if no m exists.

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A173574 4-Factorions: equal to the sum of the quadruple factorials of their digits in base 10.

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Maple

Mathematica

A173575 5-Factorions: equal to the sum of the quintuple factorials of their digits in base 10.

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Maple

A173576 6-Factorions: equal to the sum of the 6-ple factorials of their digits in base 10.

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Maple

A188284 Finite sequence of numbers n such that iterations for the map r -> A061602(r) starting with n ends with the same number n.

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A303935 Size of orbit of n under repeated application of sum of factorial of digits of n.

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Mathematica

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A101698 Numbers n such that n = d_1*d_1!+d_2*d_2!+...+d_k*d_k!, where d_1 d_2 ... d_k is the decimal expansion of n.

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Mathematica

A163576 Near-factorions: equal to the sum of the factorials of all but one of their digits in base 10.

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MATLAB

A247227 Numbers that divide the sum of the factorials of their digits in base 10.

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Magma

Mathematica

PARI

A101698 Numbers n such that n = d_1d_1!+d_2d_2!+...+d_k*d_k!, where d_1 d_2 ... d_k is the decimal expansion of n.