A066459 -id:A066459 - cp's OEIS frontend

A066457 Numbers k such that product of factorials of digits of k equals pi(k) (A000720).

13, 1512, 1520, 1521, 12016, 12035, 226130351, 209210612202, 209210612212, 209210612220, 209210612221, 13030323000581525

Offset: 1

Jason Earls, Jan 02 2002

The Caldwell/Honaker paper does not discuss this, only suggests further areas of investigation.
There are no other members of the sequence up to and including n=1000000. - Harvey P. Dale, Jan 07 2002
If 10n is in the sequence and 10n+1 is composite then 10n+1 is also in the sequence (the proof is easy). - Farideh Firoozbakht, Oct 24 2008
a(13) > 10^19 if it exists. - Chai Wah Wu, May 03 2018

			12016 is a term because there are exactly 1!*2!*0!*1!*6! (or 1440) prime numbers less than or equal to 12016.
pi(209210612202) = 8360755200 = 2!*0!*9!*2!*1!*0!*6!*1!*2!*2!*0!*2!. [Qu,Shun Liang (medie2006(AT)126.com), Nov 23 2008]

C. Caldwell and G. L. Honaker, Jr., Is pi(6521)=6!+5!+2!+1! unique?
A discussion about this topic: bbs.emath.ac.cn [From Qu,Shun Liang (medie2006(AT)126.com), Nov 23 2008]
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.

Cf. A000720, A066459, A049529, A105327.

Select[Range[1000000], Times@@( # !&/@IntegerDigits[ # ])==PrimePi[ # ]&]

isok(n) = my(d = digits(n)); prod(k=1, #d, d[k]!) == primepi(n); \\ Michel Marcus, May 04 2018

a(7) from Farideh Firoozbakht, Apr 20 2005
a(8)-a(11) from Qu,Shun Liang (medie2006(AT)126.com), Nov 23 2008
a(12) from Chai Wah Wu, May 03 2018

A197181 Numbers that are a divisor of the product of the factorials of their digits in decimal representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 16, 18, 24, 28, 36, 45, 48, 60, 64, 70, 72, 75, 80, 84, 90, 96, 98, 128, 144, 168, 175, 180, 189, 192, 256, 280, 288, 360, 378, 384, 448, 480, 486, 567, 576, 588, 640, 648, 672, 675, 720, 729, 756, 768, 784, 840, 864, 875, 882

Offset: 1

Reinhard Zumkeller, Oct 11 2011

			a(10)=15: A066459(15) = 1!*5! = 120 = 15 * 8;
a(11)=16: A066459(16) = 1!*6! = 720 = 16 * 45;
17 is not a term because 5040 mod 17 = 8, A066459(16) = 5040;
a(12)=18: A066459(15) = 1!*8! = 40320 = 18 * 2240.

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

Cf. A066459, A000142, A188264.

import Data.List (elemIndices)
a197181 n = a197181_list !! (n-1)
a197181_list = map (+ 1) $ elemIndices 0 $
   zipWith mod (map a066459 [1..]) [1..]

Select[Range[900],IntegerQ[Times@@(IntegerDigits[#]!)/#]&] (* Harvey P. Dale, Jul 07 2019 *)

A034879 a(n) = product of factorials of digits of a(n-1).

Original entry on oeis.org

3, 6, 720, 10080, 40320, 288, 3251404800, 33443020800, 10032906240, 150493593600, 1179600089304268800000, 6494947622660923209343932825600000000, 817033558319976619871285124179533435912396304547840000000000000

Offset: 1

Erich Friedman

The next term has 110 digits. - Harvey P. Dale, Oct 17 2011

a034879 n = a034879_list !! (n-1)
a034879_list = iterate a066459 3  -- Reinhard Zumkeller, Oct 11 2011

NestList[Times@@(IntegerDigits[#]!)&,3,12] (* Harvey P. Dale, Oct 17 2011 *)

a(n+1) = A066459(a(n)), a(1) = 3. - Reinhard Zumkeller, Oct 11 2011

a(13) from Harvey P. Dale, Oct 17 2011

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

A082939 Numbers such that sum of the digits of the product of the factorial of digits of the number is equal to the sum of the digits of the number.

Original entry on oeis.org

1, 2, 10, 18, 20, 22, 27, 36, 63, 72, 81, 100, 108, 114, 117, 126, 135, 141, 153, 162, 171, 180, 200, 202, 207, 216, 220, 261, 270, 306, 315, 333, 351, 360, 411, 513, 531, 603, 612, 621, 630, 702, 711, 720, 801, 810, 1000, 1008, 1014, 1017, 1026, 1035, 1041

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 27 2003

			63 = 6!*3! = 720*6 = 4320, 4 + 3 + 2 + 0 = 9 and 6 + 3 = 9.

Suggested by Amarnath Murthy.

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

Cf. A007953, A066459, A082940.

from math import factorial, prod
def ok(n):
    d = list(map(int, str(n)))
    return sum(map(int, str(prod(map(factorial, d))))) == sum(d)
print([k for k in range(1042) if ok(k)]) # Michael S. Branicky, Aug 15 2022

Numbers k such that A007953(k) = A007953(A066459(k)).

Corrected and extended by Jason Earls, May 22 2004

A137603 Numbers m such that product of factorials of digits of m equals sigma(m).

Original entry on oeis.org

1, 14, 1253, 2261, 2622, 13145, 13630, 20146, 24035, 30362, 31416, 42504, 50424, 63240, 112281, 117124, 126005, 150360, 161160, 225153, 252126, 262105, 318021, 341630, 510632, 611523, 723104, 1071521, 1131190, 1153262, 1200626, 1242108

Offset: 1

Farideh Firoozbakht, Apr 11 2008

			sigma(10230248)=1!*0!*2!*3!*0!*2!*4!*8! so 10230248 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A066457, A066459, A000203.

Select[Range[1250000],Times@@(IntegerDigits[#]!)==DivisorSigma[1,#]&] (* James C. McMahon, Jun 01 2025 *)

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

Original entry on oeis.org

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 *)

A384955 a(n) is the multinomial coefficient of the digits of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005

Offset: 0

Stefano Spezia, Jun 13 2025

			a(35) = (3+5)!/(3!*5!) = 40320/(6*120) = 56;
a(1512) = (1+5+1+2)!/(1!*5!*1!*2!) = 362880/(120*2) = 1512.

Stefano Spezia, Table of n, a(n) for n = 0..10000

Cf. A037124, A066459, A093659, A269221.

a:= n-> (l-> combinat[multinomial](add(i,i=l), l[]))(convert(n, base, 10)):
seq(a(n), n=0..69);  # Alois P. Heinz, Jun 15 2025

a[n_]:=Multinomial @@IntegerDigits[n]; Array[a,70,0]

from math import factorial, prod
def a(n): return factorial(sum(digits:=list(map(int, str(n))))) // prod(factorial(x) for x in digits)
print([a(n) for n in range(70)]) # David Radcliffe, Jun 15 2025

a(n) = A269221(n)/A066459(n).
a(n) = 1 iff n is equal to 0 or has only one nonzero digit (cf. A037124).
Conjecture: a(n) = n iff n = 1 or n = 1512.

cp's OEIS Frontend

A066457 Numbers k such that product of factorials of digits of k equals pi(k) (A000720).

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A197181 Numbers that are a divisor of the product of the factorials of their digits in decimal representation.

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A034879 a(n) = product of factorials of digits of a(n-1).

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

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A082939 Numbers such that sum of the digits of the product of the factorial of digits of the number is equal to the sum of the digits of the number.

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A137603 Numbers m such that product of factorials of digits of m equals sigma(m).

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

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A384955 a(n) is the multinomial coefficient of the digits of n.

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