A072124 -id:A072124

A072177

a(n)-th factorial is the smallest factorial containing exactly n 3's, or 0 if no such number exists.

+10
9

8, 15, 25, 36, 24, 49, 32, 54, 43, 69, 76, 89, 84, 113, 82, 105, 112, 92, 114, 106, 118, 107, 109, 151, 166, 143, 160, 149, 190, 152, 158, 172, 176, 0, 192, 181, 183, 177, 180, 202, 200, 193, 226, 238, 242, 223, 251, 227, 290, 261, 267, 292, 265, 300, 295, 285

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

It is conjectured that a(34)=0 since no factorial < 10000 contained just 34 threes.

The 500-term b-file contains 16 zeros, each relying on the same conjecture, i.e., that because there is no factorial < 10000! containing just n threes no factorial satisfies the condition. - Harvey P. Dale, Jan 02 2021

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..500

EXAMPLE

a(2)=15 since the 15th factorial, i.e., 15!=1307674368000, contains exactly two 3's.

MATHEMATICA

Do[k = 1; While[ Count[IntegerDigits[k! ], 3] != n, k++ ]; Print[k], {n, 1, 60}]

With[{fc=Range[400]!}, Table[Position[fc, _?(DigitCount[#, 10, 3]==n&), 1, 1]/.{}->0, {n, 60}]]//Flatten (* Harvey P. Dale, Jan 02 2021 *)

CROSSREFS

Cf. A072269, A072220, A072208, A072204, A072200, A072199, A072178, A072163 & A072124.

KEYWORD

base,nonn

AUTHOR

Shyam Sunder Gupta, Jul 30 2002

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jul 31 2002

STATUS

approved

A072204

a(n) is the smallest integer k whose factorial contains exactly n 7's, or 0 if no such number exists.

+10
9

6, 14, 18, 31, 49, 22, 54, 48, 56, 71, 82, 72, 86, 81, 92, 97, 87, 122, 91, 119, 131, 112, 121, 140, 104, 152, 144, 173, 127, 157, 172, 201, 227, 179, 200, 187, 183, 210, 236, 221, 193, 217, 279, 212, 213, 235, 238, 289, 265, 228, 256, 261, 250, 242, 285, 307

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Michael S. Branicky, Table of n, a(n) for n = 1..148

Michael S. Branicky, Table of n, a(n) for n = 0..1000 with presumed 0 values listed (search bound 93000)

EXAMPLE

a(2) = 14 since the 14th factorial, i.e., 14! = 87178291200, contains exactly two 7's.

MATHEMATICA

Do[k = 1; While[ Count[IntegerDigits[k! ], 7] != n, k++ ]; Print[k], {n, 1, 60}]

PROG

(Python)

def a(n, bound=10**4):

k = f = 1

while not str(f).count('7') == n and k <= bound: k += 1; f *= k

return k * (k <= bound)

print([a(n) for n in range(1, 149)]) # Michael S. Branicky, Jun 14 2021

CROSSREFS

Cf. A072238, A072220, A072208, A072200, A072199, A072178, A072177, A072163, A072124.

KEYWORD

base,nonn

AUTHOR

Shyam Sunder Gupta, Jul 30 2002

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jul 31 2002

STATUS

approved

A072220

a(n)-th factorial is the smallest factorial containing exactly n 9's, or 0 if no such number exists.

+10
9

12, 11, 21, 29, 34, 46, 36, 59, 79, 75, 0, 70, 82, 90, 95, 97, 112, 89, 105, 96, 134, 130, 127, 165, 142, 149, 144, 145, 161, 163, 182, 189, 160, 178, 139, 180, 206, 192, 224, 214, 188, 215, 226, 207, 218, 267, 283, 261, 268, 262, 240, 280, 234, 285, 343, 277, 284, 269, 281, 331, 278, 308

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

It is conjectured that a(11) = 0 since no factorial < 10000 contains exactly eleven nines.

LINKS

Table of n, a(n) for n=1..62.

Robert Israel, Table of n, a(n) for n = 1 .. 1000 (entries of 0 are conjectured)

EXAMPLE

a(2) = 11 since 11! = 39916800 contains exactly two 9's.

MAPLE

f:= n -> numboccur(9, convert(n, base, 10)):

V:= Vector(100):

q:= 1:

for i from 2 to 2000 do

q:= i*q; v:= f(q);

if v > 0 and v < 100 and V[v] = 0 then V[v]:= i; fi

od:

convert(V, list); # Robert Israel, Sep 29 2024

MATHEMATICA

Do[k = 1; While[ Count[IntegerDigits[k! ], 9] != n, k++ ]; Print[k], {n, 1, 60}]

Module[{c=Table[{n, DigitCount[n!, 10, 9]}, {n, 350}]}, Table[SelectFirst[c, #[[2]]==m&], {m, 60}]][[;; , 1]]/."NotFound"->0 (* Harvey P. Dale, Sep 18 2023 *)

PROG

(Python)

def a(n, multiple_limit=300):

fk, limit = 1, multiple_limit*n

for k in range(1, limit+1):

fk *= k

if str(fk).count("9") == n: return k

return 0

print([a(n) for n in range(1, 57)]) # Michael S. Branicky, Dec 11 2021

CROSSREFS

Cf. A072232, A072208, A072204, A072200, A072199, A072178, A072177, A072163, A072124.

KEYWORD

base,nonn

AUTHOR

Shyam Sunder Gupta, Jul 30 2002

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jul 31 2002

More terms from Robert Israel, Sep 29 2024

STATUS

approved

A072163

a(n)-th factorial is the smallest factorial containing exactly n 2's, or 0 if no such number exists.

+10
8

2, 14, 13, 30, 40, 47, 31, 46, 54, 49, 65, 76, 62, 69, 107, 78, 86, 115, 95, 121, 109, 165, 110, 113, 149, 151, 146, 137, 162, 159, 170, 191, 195, 174, 190, 196, 164, 209, 202, 173, 198, 248, 201, 262, 231, 205, 263, 233, 246, 256, 270, 244, 287, 200, 271, 250

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..56.

EXAMPLE

a(2)=14 since 14th factorial, i.e., 14!=87178291200, contains exactly two 2's.

MATHEMATICA

Do[k = 1; While[ Count[IntegerDigits[k! ], 2] != n, k++ ]; Print[k], {n, 1, 60}]

CROSSREFS

Cf. A072293, A072220, A072208, A072204, A072200, A072199, A072178, A072177, A072124.

KEYWORD

base,nonn

AUTHOR

Shyam Sunder Gupta, Jul 30 2002

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jul 31 2002

STATUS

approved

A072178

a(n)-th factorial is the smallest factorial containing exactly n 4's, or 0 if no such number exists.

+10
8

4, 19, 21, 24, 44, 42, 50, 57, 0, 60, 76, 91, 56, 86, 85, 66, 92, 88, 114, 129, 131, 106, 130, 122, 117, 157, 134, 175, 119, 150, 181, 165, 185, 179, 198, 182, 220, 228, 188, 190, 261, 235, 222, 231, 229, 233, 224, 227, 288, 372, 241, 279, 254, 253, 318, 267

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

It is conjectured that a(9) = 0 since no factorial < 10000 contained just 9 fours.

LINKS

Table of n, a(n) for n=1..56.

EXAMPLE

a(2) = 19 since 19th factorial, i.e., 19! = 121645100408832000 contains exactly two 4's.

MATHEMATICA

Do[k = 1; While[ Count[IntegerDigits[k! ], 4] != n, k++ ]; Print[k], {n, 1, 60}]

PROG

(Python)

from math import factorial

def a(n, limit=1000):

fact = 1

for t in range(limit+1):

if str(fact).count('4') == n: return t

fact *= (t+1)

return 0

print([a(n) for n in range(1, 57)]) # Michael S. Branicky, Apr 19 2021

CROSSREFS

Cf. A072244, A072220, A072208, A072204, A072200, A072199, A072177, A072163, A072124.

KEYWORD

base,nonn

AUTHOR

Shyam Sunder Gupta, Jul 30 2002

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jul 31 2002

STATUS

approved

A072199

a(n)-th factorial is the smallest factorial containing exactly n 5's, or 0 if no such number exists.

+10
8

7, 17, 25, 30, 37, 41, 43, 46, 75, 65, 55, 83, 74, 80, 94, 116, 91, 114, 131, 111, 115, 136, 147, 125, 128, 143, 102, 169, 152, 157, 197, 150, 165, 185, 193, 190, 206, 198, 192, 214, 236, 203, 242, 226, 205, 256, 251, 220, 270, 239, 230, 261, 286, 222, 264

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..55.

EXAMPLE

a(2)=17 since 17th factorial, i.e., 17!=355687428096000 contains exactly two 5's.

MATHEMATICA

Do[k = 1; While[ Count[IntegerDigits[k! ], 5] != n, k++ ]; Print[k], {n, 1, 60}]

With[{fs=Table[{n, n!}, {n, 500}]}, Table[SelectFirst[fs, DigitCount[#[[2]], 10, 5] == k&], {k, 100}]][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 16 2018 *)

CROSSREFS

Cf. A072242, A072220, A072208, A072204, A072200, A072178, A072177, A072163 & A072124.

KEYWORD

base,nonn

AUTHOR

Shyam Sunder Gupta, Jul 30 2002

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jul 31 2002

STATUS

approved

A072200

a(n)-th factorial is the smallest factorial containing exactly n 6's, or 0 if no such number exists.

+10
8

3, 15, 23, 26, 32, 41, 35, 45, 50, 72, 63, 83, 84, 98, 89, 94, 91, 121, 99, 142, 117, 160, 129, 0, 127, 131, 132, 154, 153, 163, 170, 179, 190, 178, 166, 189, 217, 209, 206, 174, 208, 199, 207, 211, 214, 245, 263, 175, 240, 255, 295, 234, 213, 296, 286, 266, 278

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

It is conjectured that a(24)=0 since no factorial < 10000 contained just 24 sixes.

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

a(2)=15 since the 15th factorial, i.e., 15!=1307674368000, contains exactly two 6's.

MATHEMATICA

Do[k = 1; While[ Count[IntegerDigits[k! ], 6] != n, k++ ]; Print[k], {n, 1, 60}]

CROSSREFS

Cf. A072240, A072220, A072208, A072204, A072199, A072178, A072177, A072163 & A072124.

KEYWORD

base,nonn

AUTHOR

Shyam Sunder Gupta, Jul 30 2002

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jul 31 2002

STATUS

approved

A072208

a(n)-th factorial is the smallest factorial containing exactly n 8's, or 0 if no such number exists.

+10
8

11, 9, 36, 16, 30, 27, 39, 33, 44, 58, 56, 64, 80, 70, 72, 94, 97, 71, 108, 143, 120, 134, 118, 162, 125, 133, 151, 137, 138, 159, 169, 197, 184, 171, 178, 176, 206, 177, 191, 208, 207, 240, 252, 232, 239, 270, 229, 308, 243, 223, 278, 257, 250, 0, 303, 242, 311

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

It is conjectured that a(54)=0 since no factorial < 10000 contained just 54 eights.

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

a(2)=9 since 9th factorial i.e. 9!=362880 contains exactly two 8's.

MATHEMATICA

Do[k = 1; While[ Count[IntegerDigits[k! ], 8] != n, k++ ]; Print[k], {n, 1, 60}]

Transpose[Flatten[Table[Select[Table[{n, DigitCount[n!, 10, 8]}, {n, 500}], Last[#] == i&, 1], {i, 50}], 1]][[1]] (* Harvey P. Dale, Sep 13 2013 *)

CROSSREFS

Cf. A072237, A072220, A072204, A072200, A072199, A072178, A072177, A072163, A072124.

KEYWORD

base,nonn

AUTHOR

Shyam Sunder Gupta, Jul 30 2002

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jul 31 2002

STATUS

approved

A072295

Smallest factorial containing exactly n 1's.

+10
2

1, 87178291200, 121645100408832000, 15511210043330985984000000, 263130836933693530167218012160000000, 815915283247897734345611269596115894272000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..83

FORMULA

a(n) = A000142(A072124(n)). - Amiram Eldar, Sep 01 2020

EXAMPLE

a(2) = 87178291200 since 87178291200 is the smallest factorial containing exactly two 1's.

MATHEMATICA

With[{fctrs=Range[100]!}, Table[First[Select[fctrs, DigitCount[#, 10, 1] == n&]], {n, 10}]] (* Harvey P. Dale, Sep 29 2011 *)

CROSSREFS

Cf. A000142, A072124.

KEYWORD

base,nonn

AUTHOR

Shyam Sunder Gupta, Jul 30 2002

STATUS

approved