A061602 -id:A061602 - cp's OEIS frontend

A254499 Amicable factorions.

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).

A052279 Primes such that the sum of the factorials of the digits is a perfect square.

Original entry on oeis.org

17, 41, 71, 211, 433, 457, 547, 1013, 1031, 1103, 1301, 1489, 2063, 3001, 3011, 4451, 5077, 5441, 5651, 5717, 6203, 6551, 7057, 7507, 7517, 8419, 8941, 10163, 10613, 10631, 16103, 16301, 20369, 20639, 20693, 20873, 20963, 21313, 21661, 23003, 23087, 23131

Offset: 1

G. L. Honaker, Jr., Feb 05 2000

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A000142, A000290, A061602.

Select[Prime[Range[3000]],IntegerQ[Sqrt[Total[IntegerDigits[#]!]]]&] (* Harvey P. Dale, Apr 11 2016 *)

isok(n) = isprime(n) && (d = digits(n)) && issquare(sum(i=1, #d, d[i]!)); \\ Michel Marcus, Jan 06 2014

More terms from Michel Marcus, Jan 06 2014

A093325 Numbers that are divisible by the sum of the factorials of their digits.

Original entry on oeis.org

1, 2, 10, 12, 21, 32, 104, 111, 112, 120, 145, 200, 220, 222, 224, 341, 403, 441, 1000, 1020, 1100, 1120, 1122, 1200, 1204, 1210, 1212, 1230, 1232, 1320, 1330, 2000, 2010, 2030, 2100, 2110, 2112, 2123, 2125, 2130, 2204, 2212, 2232, 2250, 2300, 2310, 2321

Offset: 1

Jason Earls, May 11 2004

			104 is a term because 1!+0!+4!=26 and 104/26=4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005349, A061602.

f:=func< n|&+[Factorial(Intseq(n)[i]):i in [1..#Intseq(n)]]>; [k:k in [1..2500]| k mod f(k) eq 0]; // Marius A. Burtea, Dec 16 2019

seqQ[n_] := Divisible[n, Total@(Factorial /@ IntegerDigits[n])]; Select[Range[2500], seqQ] (* Amiram Eldar, Dec 16 2019 *)

A109016 Concatenate n and the sum of factorials of the digits of n.

Original entry on oeis.org

1, 11, 22, 36, 424, 5120, 6720, 75040, 840320, 9362880, 102, 112, 123, 137, 1425, 15121, 16721, 175041, 1840321, 19362881, 203, 213, 224, 238, 2426, 25122, 26722, 275042, 2840322, 29362882, 307, 317, 328, 3312, 3430, 35126, 36726, 375046

Offset: 0

Jason Earls, Jun 16 2005

			a(16)=16721 because 1!+6! = 721.

Cf. A061602.

Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[Total[IntegerDigits[n]!]]]],{n,0,40}] (* Harvey P. Dale, Jul 13 2023 *)

A111095 n = Sum_{b} c_b*b! in the factorial base rewritten by c_b-fold repetition of b, b=1,2,3,....

Original entry on oeis.org

1, 2, 12, 22, 122, 3, 13, 23, 123, 223, 1223, 33, 133, 233, 1233, 2233, 12233, 333, 1333, 2333, 12333, 22333, 122333, 4, 14, 24, 124, 224, 1224, 34, 134, 234, 1234, 2234, 12234, 334, 1334, 2334, 12334, 22334

Offset: 1

Giorgio Balzarotti and Paolo P. Lava, Oct 13 2005

The integer n has a unique "greedy" representation in the factorial base as n = Sum_{b>=1} c_b*b!, see A007623.
The number of coefficients c_b is A084558(n).
The current sequence starts from an empty string, scans the coefficients c_b in the order b=1,2,3,..., i.e., reads A007623(n) from the least to the most significant position, and appends b c_b times to the string. The resulting string is shown in the sequence as a standard decimal number a(n).

			a(39) = 12334 with A007623(39) = 1211, because 1! + 2! + 3! + 3! + 4! = 1 + 2 + 6 + 6 + 24 = 39

Cf. A005096, A007623, A108911.

A061602(a(n)) = n. - R. J. Mathar, Oct 30 2010

Definition and comment shortened with reference to A007623 - R. J. Mathar, Oct 30 2010

A242868 Numbers n such that sum of the factorial of digits of n is semiprime.

Original entry on oeis.org

3, 14, 15, 16, 17, 18, 22, 24, 25, 27, 28, 40, 41, 42, 50, 51, 52, 60, 61, 70, 71, 72, 80, 81, 82, 102, 104, 105, 107, 108, 112, 114, 115, 117, 118, 120, 121, 123, 125, 126, 128, 132, 140, 141, 144, 145, 146, 147, 148, 150, 151, 152, 154, 156, 157, 158, 162, 164

Offset: 1

K. D. Bajpai, May 24 2014

			a(4) = 16:  1! + 6! = 1 + 720 = 721 = 7 * 103 which is semiprime.
a(9) = 25:  2! + 5! = 2 + 120 = 122 = 2 * 61 which is semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001358, A061602, A165451.

with(numtheory):A242868:= proc() local a; a:=add( i!,i = convert((n), base, 10))(n);if bigomega(a)=2 then RETURN (n);fi; end: seq(A242868 (), n=1..300);

Select[Range[200],PrimeOmega[Total[IntegerDigits[#]!]]==2&] (* Harvey P. Dale, Jul 30 2015 *)

A061603 a(n) = n! / {product of factorials of the digits of n}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3628800, 39916800, 239500800, 1037836800, 3632428800, 10897286400, 29059430400, 70572902400, 158789030400, 335221286400, 1216451004088320000, 25545471085854720000, 281000181944401920000, 2154334728240414720000

Offset: 0

Amarnath Murthy, May 19 2001

It can be shown that the terms obtained by the above formula are positive integers using the fact that k! divides a product of k consecutive numbers.

			a(12) = (12!) / (1!*2!) = 239500800.

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A000142, A061602, A066459.

Table[n!/Times@@(IntegerDigits[n]!),{n,0,30}] (* Harvey P. Dale, Jan 19 2017 *)

a(n) = my(d = digits(n)); n!/prod(k=1, #d, d[k]!); \\ Michel Marcus, Jul 02 2018

a(n) = A000142(n)/A066459(n). - Michel Marcus, Jul 02 2018

Corrected and extended by Vladeta Jovovic, May 19 2001

A074257 Sum of factorials of digits of n equals the largest prime factor of n.

Original entry on oeis.org

2, 12, 341, 403, 1200, 1232, 2000, 2204, 4530, 4614, 5134, 10000, 13200, 13345, 14210, 21141, 23100, 31433, 40020, 101442, 111252, 111452, 112000, 112320, 123201, 135453, 145343, 162121, 173434, 200025, 202106, 203050, 210000, 211420

Offset: 1

Jason Earls, Sep 20 2002

			4614 = 2*3*769 and 4!+6!+1!+4! = 769.

Vincenzo Librandi and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 250 terms from Librandi)

Cf. A061602.

Select[Range[220000],FactorInteger[#][[-1,1]]==Total[IntegerDigits[#]!]&] (* Harvey P. Dale, Mar 08 2014 *)

gpf(n)=if(n<2,1,my(f=factor(n)[,1]);f[#f])
is(n)=my(d=digits(n),t=sum(i=1,#d,d[i]!));n%t==0&&isprime(t)&&gpf(n/t)<=t \\ Charles R Greathouse IV, Mar 10 2014

A084405 Primes whose sum of factorials of digits is also prime.

Original entry on oeis.org

2, 11, 13, 31, 101, 163, 313, 331, 431, 503, 613, 631, 1021, 1201, 1223, 1433, 1439, 1453, 1483, 1493, 1543, 1567, 1657, 1663, 1667, 1669, 1753, 1777, 1789, 1879, 1987, 1999, 2011, 2111, 2203, 2213, 2221, 3049, 3163, 3221, 3313, 3331, 3361, 3413, 3461, 3491

Offset: 1

Jason Earls, Jun 24 2003

			a(10)=503, a prime, and 5! + 0! + 3! = 127, a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A061602.

Select[Prime[Range[500]],PrimeQ[Total[IntegerDigits[#]!]]&] (* Harvey P. Dale, Mar 20 2016 *)

{digitsumfac(n)=local(s, d); s=0; while(n>0,d=divrem(n,10); n=d[1]; s=s+d[2]!); s}
{facp(m)=local(ct,sr); ct=0; sr=0; forprime(p=2,m, if(isprime(digitsumfac(p)),ct++; print1(p," "); sr+=(1.0/p); )); print(); print("Found: "ct" primes < "m); print("Sum of reciprocals = "sr); }

from sympy import isprime
from math import factorial
def f(n): return sum(factorial(int(d)) for d in str(n))
def ok(n): return isprime(n) and isprime(f(n))
print([k for k in range(3500) if ok(k)]) # Michael S. Branicky, Feb 11 2023

A130687 Numbers n such that a_1! + a_2! + ... + a_m! is a square number, where a_1a_2...a_m is the decimal expansion of n.

Original entry on oeis.org

1, 14, 15, 17, 22, 40, 41, 45, 50, 51, 54, 70, 71, 102, 112, 120, 121, 123, 132, 144, 156, 165, 200, 201, 203, 210, 211, 213, 230, 231, 302, 312, 320, 321, 334, 343, 404, 414, 433, 440, 441, 457, 475, 506, 516, 547, 560, 561, 574, 605, 615

Offset: 1

Yalcin Aktar, Jun 30 2007

			1! + 4! = 4! + 1! = 5^2, hence 14 and 41 are in the sequence.

A061602 := proc(n) local digs ; digs := convert(n,base,10) ; add(factorial(op(i,digs)),i=1..nops(digs)) ; end: isA130687 := proc(n) issqr(A061602(n)) ; end: for n from 1 to 3000 do if isA130687(n) then printf("%d, ",n) ; fi ; od ; # R. J. Mathar, Jul 12 2007

Select[Range[755], IntegerQ[Sqrt[DigitCount[ # ][[10]]+Sum[DigitCount[ # ][[i]]*i!, {i, 1, 9}]]] &]

A010052(A061602(a(n)))=1. - R. J. Mathar, Jul 12 2007

Edited by Stefan Steinerberger and R. J. Mathar, Jul 12 2007

cp's OEIS Frontend

A254499 Amicable factorions.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A052279 Primes such that the sum of the factorials of the digits is a perfect square.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A093325 Numbers that are divisible by the sum of the factorials of their digits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A109016 Concatenate n and the sum of factorials of the digits of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A111095 n = Sum_{b} c_b*b! in the factorial base rewritten by c_b-fold repetition of b, b=1,2,3,....

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A242868 Numbers n such that sum of the factorial of digits of n is semiprime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A061603 a(n) = n! / {product of factorials of the digits of n}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A074257 Sum of factorials of digits of n equals the largest prime factor of n.

Views

Author

Keywords

Examples

Links

Crossrefs