A061602 -id:A061602 - cp's OEIS frontend

A188264 Numbers m that are divisible by the product of the factorials of their digits in base 10.

1, 2, 10, 11, 12, 20, 30, 100, 101, 102, 110, 111, 112, 120, 132, 200, 210, 212, 220, 240, 300, 312, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1032, 1100, 1101, 1102, 1104, 1110, 1111, 1112, 1120, 1200, 1210, 1212, 1220, 1320, 2000, 2010, 2012, 2020, 2100, 2110, 2112

Offset: 1

Jaroslav Krizek, Mar 25 2011

			Number 30 is in sequence because 30 is divisible by the product of factorials 3!*0! = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A061602, A066459, A093325.

Cf. A066459, A000142, A197181.

import Data.List (elemIndices)
a188264 n = a188264_list !! (n-1)
a188264_list =
   map (+ 1) $ elemIndices 0 $ zipWith mod [1..] $ map a066459 [1..]
-- Reinhard Zumkeller, Oct 11 2011

Select[Range[2200],Divisible[#,Times@@(IntegerDigits[#]!)]&] (* Harvey P. Dale, May 24 2017 *)

A242855 Catalan numbers C(n) such that sum of the factorials of digits of C(n) is prime.

Original entry on oeis.org

2, 16796, 263747951750360, 1002242216651368, 104088460289122304033498318812080, 22033725021956517463358552614056949950, 1000134600800354781929399250536541864362461089950800, 216489185503133990863274261791925599831188392742851863147080

Offset: 1

K. D. Bajpai, May 24 2014

The n-th Catalan number C(n) = (2*n)!/(n!*(n+1)!).
The next term, a(9), has 66 digits which is too large to display in data section.
The 102nd term, a(102), having 992 digits, is the last term in b-file.
a(103) has 1021 digits, hence not included in b-file.
Intersection of A000108 and A165451.

			16796 = (2*10)!/(10!*(10+1)!) is 10th Catalan number: 1!+6!+7!+9!+6! = 369361 which is prime.
263747951750360 = (2*28)!/(28!*(28+1)!) is 28th Catalan number: 2!+6!+3!+7!+4!+7!+9!+5!+1!+7!+5!+0!+3!+6!+0! = 379721 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..102

Cf. A000040, A000108, A061602, A165451.

with(numtheory):A242855:= proc() if isprime(add( i!,i = convert(((2*n)!/(n!*(n+1)!)), base, 10))((2*n)!/(n!*(n+1)!))) then RETURN ((2*n)!/(n!*(n+1)!)); fi; end: seq(A242855 (), n=1..50);

Select[CatalanNumber[Range[150]],PrimeQ[Total[IntegerDigits[#]!]]&] (* Harvey P. Dale, Apr 30 2025 *)

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))

A108257 Numbers k such that concatenating k and the sum of factorials of the digits of k produces a prime.

Original entry on oeis.org

1, 13, 15, 30, 31, 91, 101, 110, 128, 133, 136, 138, 144, 152, 156, 166, 175, 193, 199, 203, 215, 230, 250, 260, 280, 281, 303, 304, 306, 307, 309, 315, 320, 330, 331, 340, 361, 391, 412, 508, 520, 550, 606, 651, 661, 681, 708, 712, 717, 730, 750, 751, 780

Offset: 1

Jason Earls, Jun 18 2005

The largest prime I have found pertaining to this sequence is A109016(Fibonacci(9837)) with 2064 digits (not proved prime, only Fermat and Lucas PRP).

			193 is in the sequence because 1!+9!+3! = 362887 and 193362887 is prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A061602, A109016.

Select[Range[780],PrimeQ[FromDigits[Join[IntegerDigits[#],IntegerDigits[Total[IntegerDigits[#]!]]]]]&] (* James C. McMahon, Feb 22 2024 *)

from math import factorial
from sympy import isprime
def ok(n):
    return isprime(int((s:=str(n))+str(sum(factorial(int(d)) for d in s))))
print([k for k in range(999) if ok(k)]) # Michael S. Branicky, Feb 22 2024

A121103 a(1)=1; a(n) = the reversal of (a(n-1) + sfd(a(n-1))), where sfd(n)= sum of factorials of the digits of n.

Original entry on oeis.org

1, 2, 4, 82, 40404, 87404, 318231, 765853, 971218, 2649731, 4048103, 848804, 318969, 6775801, 3407286, 9933543, 56495601, 78106865, 70149187, 49426507, 81359794, 56013128, 99245065, 21817999, 30025922, 63988303, 26523446

Offset: 1

Jason Earls, Aug 12 2006

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

Cf. A061602.

NestList[IntegerReverse[#+Total[IntegerDigits[#]!]]&,1,30] (* Harvey P. Dale, May 05 2023 *)

A161178 Sum of the double factorials of the digits of n.

Original entry on oeis.org

1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 2, 2, 3, 4, 9, 16, 49, 106, 385, 946, 3, 3, 4, 5, 10, 17, 50, 107, 386, 947, 4, 4, 5, 6, 11, 18, 51, 108, 387, 948, 9, 9, 10, 11, 16, 23, 56, 113, 392, 953, 16, 16, 17, 18, 23, 30, 63, 120, 399, 960, 49, 49, 50, 51, 56, 63, 96, 153, 432, 993

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jun 05 2009

			a(24) = (2!!) + (4!!) = 2 + 8 = 10. a(35) = (3!!) + (5!!) = 3 + 15 = 18.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A061602, A006882.

A161178 := proc(n) if n = 0 then 1; else add(doublefactorial(d),d=convert(n,base,10)) ; end if; end proc:

Total[IntegerDigits[#]!!]&/@Range[0,70] (* Harvey P. Dale, Aug 28 2013 *)

A164955 Sequence obtained from Fibonacci numbers by taking the factorials of each digit and summing.

Original entry on oeis.org

1, 1, 1, 2, 6, 120, 40320, 7, 3, 30, 240, 403200, 49, 14, 10086, 722, 408240, 368041, 40466, 40346, 6600, 363626, 10083, 46202, 41790, 5283, 362896, 403946, 45369, 363029, 40354, 364353, 408250, 45632, 90843, 368788, 363040, 50548, 807128, 404792, 281, 41308

Offset: 0

Parthasarathy Nambi, Sep 01 2009

There seem to be very few primes in this sequence.

			a(30) = 8!+3!+2!+0!+4!+0! = 40354 because Fibonacci(30) = 832040.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A000045, A000142, A004090, A061602.

a:= n-> add(i!, i=convert((<<0|1>, <1|1>>^n)[1,2], base, 10)):
seq(a(n), n=0..42);  # Alois P. Heinz, Jul 09 2023

Total[IntegerDigits[#]!]&/@Fibonacci[Range[0,40]] (* Harvey P. Dale, May 03 2011 *)

a(n) = A061602(A000045(n)). - Alois P. Heinz, Jul 09 2023

Offset corrected and more terms from Alois P. Heinz, Jul 09 2023

A182287 If n = p10^i + q10^(i-1) + r10^(i-2) + ... in decimal notation, then a(n) = p!10^i + q!10^(i-1) + r!10^(i-2)+ ... .

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 11, 11, 12, 16, 34, 130, 730, 5050, 40330, 362890, 21, 21, 22, 26, 44, 140, 740, 5060, 40340, 362900, 61, 61, 62, 66, 84, 180, 780, 5100, 40380, 362940, 241, 241, 242, 246, 264, 360, 960, 5280, 40560, 363120

Offset: 0

Renzo Remotti, Apr 23 2012

			a(1)=1 because 1!*10^0=1, a(15)=130 because 1!*10^1+5!*10^0=130.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Renzo Remotti, Positional Factorial Sequence

Cf. A007623, A061602.

[n eq 0 select 1 else &+[Factorial(Reverse(Intseq(n))[k])*10^(#Intseq(n)-k): k in [1..#Intseq(n)]]: n in [0..50]]; // Bruno Berselli, May 15 2012

Offset changed from 1 to 0 by Bruno Berselli, May 16 2012

A241404 Sum of n and the sum of the factorials of its digits.

Original entry on oeis.org

2, 4, 9, 28, 125, 726, 5047, 40328, 362889, 12, 13, 15, 20, 39, 136, 737, 5058, 40339, 362900, 23, 24, 26, 31, 50, 147, 748, 5069, 40350, 362911, 37, 38, 40, 45, 64, 161, 762, 5083, 40364, 362925, 65, 66, 68, 73, 92, 189, 790, 5111, 40392, 362953, 171, 172, 174

Offset: 1

Vincenzo Librandi, Apr 21 2014

			a(8) = 40328 because we have 8 + 8! = 40328.

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

Cf. A061602, A108911.

f[n_]:= n + Plus@@Factorial/@IntegerDigits[n]; Table[f[n], {n, 50}]

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

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

A242571 Triangular numbers T such that sum of the factorials of digits of T is semiprime.

Original entry on oeis.org

3, 15, 28, 105, 120, 171, 210, 231, 406, 561, 741, 820, 990, 1081, 1275, 1378, 1485, 1540, 1596, 1953, 2211, 2485, 2775, 3003, 3240, 3321, 3741, 3916, 4005, 4371, 4560, 4851, 5460, 6105, 6903, 7381, 7750, 8001, 8128, 8515, 9316, 9591, 9730, 10153, 10440, 10878

Offset: 1

K. D. Bajpai, May 24 2014

The n-th triangular number T(n) = n * (n+1)/2.

Intersection of A000217 and A242868.

			18*(18+1)/2 = 171 is triangular number. 1! + 7! + 1! = 5042 = 2 * 2521 is semiprime. Hence 171 is in the sequence.
28*(28+1)/2 = 406 is triangular number. 4! + 0! + 6! = 745 = 5 * 149 is semiprime. Hence 406 is in the sequence.

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

Cf. A001358, A000217, A061602, A242868.

with(numtheory): A242571= proc() if bigomega(add( i!,i = convert((n*(n+1)/2), base, 10))(n*(n+1)/2))=2 then RETURN (n*(n+1)/2);fi; end: seq(A242571 (),n=1..300);

fQ[n_] := PrimeOmega[ Total[ IntegerDigits[ n (n + 1)/2]!]] == 2; s = Select[ Range@ 160, fQ@# &]; s (s + 1)/2 (* Robert G. Wilson v, May 26 2014 *)

cp's OEIS Frontend

A188264 Numbers m that are divisible by the product of the factorials of their digits in base 10.

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A242855 Catalan numbers C(n) such that sum of the factorials of digits of C(n) is prime.

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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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A108257 Numbers k such that concatenating k and the sum of factorials of the digits of k produces a prime.

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A121103 a(1)=1; a(n) = the reversal of (a(n-1) + sfd(a(n-1))), where sfd(n)= sum of factorials of the digits of n.

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A161178 Sum of the double factorials of the digits of n.

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A164955 Sequence obtained from Fibonacci numbers by taking the factorials of each digit and summing.

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A182287 If n = p*10^i + q*10^(i-1) + r*10^(i-2) + ... in decimal notation, then a(n) = p!*10^i + q!*10^(i-1) + r!*10^(i-2)+ ... .

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A182287 If n = p10^i + q10^(i-1) + r10^(i-2) + ... in decimal notation, then a(n) = p!10^i + q!10^(i-1) + r!10^(i-2)+ ... .