A076219 -id:A076219 - cp's OEIS frontend

A018799 Smallest nonnegative integer m such that m! begins with n in base 10.

0, 2, 9, 8, 7, 3, 6, 14, 96, 27, 22, 5, 15, 42, 25, 89, 69, 76, 63, 16, 87, 113, 54, 4, 23, 30, 205, 85, 34, 28, 62, 164, 41, 245, 17, 9, 36, 128, 11, 8, 185, 53, 351, 73, 369, 118, 12, 265, 129, 7, 21, 38, 235, 66, 46, 258, 81, 597, 279, 43, 72, 13, 559, 18, 203, 120, 311

Offset: 1

David W. Wilson

Record high values are m = 0, 2, 9, 14, 96, 113, 205, 245, 351, 369, 597, ... (see A279089); these occur, respectively, at n = 1, 2, 3, 8, 9, 22, 27, 34, 43, 45, 58, ... (see A279090). - Jon E. Schoenfield, Jan 30 2017

The existence of such m for each n was proven by Maxfield in 1970. The first 999 terms of this sequence were calculated by Southard in 1983. - Amiram Eldar, Dec 18 2018

			Since no factorial below 96! ~ 9.91*10^149 starts with 9, we have a(9) = 96. Similarly, 16 first appears as the leading digits of 89! ~ 1.65*10^136 and hence a(16) = 89. - _Lekraj Beedassy_, Oct 31 2010 and _Robert G. Wilson v_, Nov 05 2010

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David W. Wilson).
John E. Maxfield, A Note on N!, Mathematics Magazine, Vol. 43. No. 2 (1970), pp. 64-67.
Laura Southard, Investigations on Maxfield's Theorem, Pi Mu Epsilon Journal, Vol. 7, No. 8 (1983), pp. 493-495, alternative link.

Cf. A018854.
Apart from leading term, identical to A076219.
Cf. A000142, A008905, A279089, A279090.

f[n_] := Block[{k = 0, m}, While[ m = Max[0, Floor@ Log[10, k! ] - Floor@ Log[10, n]]; (k! - Mod[k!, 10^m])/10^m != n, k++ ]; k]; Array[f, 67] (* Robert G. Wilson v, Nov 05 2010 *)

A018799(n)={ local( F=1,k=0 ); while( F\1!=n, F*=k++; while( F>=n+1, F/=10)); k} \\ M. F. Hasler, Feb 01 2009

A155891 Least positive integer such that a(n)! starts with n, both written in binary.

Original entry on oeis.org

1, 2, 3, 7, 6, 3, 5, 20, 7, 14, 6, 4, 10, 12, 5, 20, 27, 50, 7, 14, 23, 6, 13, 4, 25, 30, 10, 12, 51, 5, 63, 24, 20, 36, 27, 50, 59, 11, 7, 14, 26, 75, 23, 9, 6, 13, 49, 71, 144, 31, 25, 30, 199, 19, 10, 46, 12, 51, 83, 5, 28, 73, 63, 62, 24, 54, 20, 36, 67, 27, 40, 74, 50, 59, 124

Offset: 1

M. F. Hasler, Feb 01 2009

			a(7) = 5 since 7 = 111_2 (i.e., written in base 2) and 5! = 1111000_2 is the least factorial to have 111 as its most significant binary digits.

Nick Hobson, Table of n, a(n) for n = 1..16383
Nick Hobson, C program.

Cf. A018799, A076219.

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See Links section.
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A155891(n)={ local( F=1,k=1 ); while( F\1!=n, F*=k++; while( F>=n+1, F/=2 )); k}

A155892 Least positive integer such that a(n)! starts with n, both written in base 3.

Original entry on oeis.org

1, 2, 12, 5, 13, 3, 19, 4, 31, 18, 12, 21, 5, 33, 17, 13, 34, 8, 38, 7, 19, 92, 24, 4, 11, 6, 46, 39, 31, 22, 18, 77, 12, 55, 32, 75, 21, 52, 74, 5, 15, 33, 61, 122, 166, 17, 47, 13, 65, 54, 49, 148, 34, 71, 8, 117, 107, 60, 38, 51, 10, 7, 111, 25, 19, 92, 123, 128, 29, 24, 56, 99

Offset: 1

M. F. Hasler, Feb 01 2009

			a(4) = 5 since 4 = 11[3] (i.e. written in base 2) and 5! = 11110[3] is the least factorial to have 11 as most significant digits in base 3.

Cf. A155891, A018799, A076219.

A155892(n)={ local( F=1,k=1 ); while( F\1!=n, F*=k++; while( F>=n+1, F/=3 )); k}

A155894 Least positive integer such that a(n)! starts with n, both written in base 16.

Original entry on oeis.org

1, 2, 10, 36, 9, 3, 5, 24, 8, 43, 82, 25, 30, 46, 64, 62, 60, 68, 7, 14, 70, 18, 13, 4, 31, 272, 19, 12, 97, 35, 161, 140, 20, 86, 27, 74, 131, 11, 502, 234, 47, 181, 386, 21, 6, 996, 52, 232, 144, 32, 393, 189, 199, 281, 10, 490, 39, 154, 83, 22, 28, 577, 63, 65, 88, 61, 115

Offset: 1

M. F. Hasler, Feb 01 2009

Nick Hobson, Table of n, a(n) for n = 1..4095
Nick Hobson, C program.

Cf. A155891, A155892, A155893, A018799, A076219.

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See Links section.
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A155894(n)={ local( F=1,k=1 ); while( F\1!=n, F*=k++; while( F>=n+1, F/=16 )); k}

A155893 Least positive integer such that a(n)! starts with n, both written in base 8.

Original entry on oeis.org

1, 2, 4, 16, 13, 3, 46, 27, 7, 14, 6, 32, 10, 39, 5, 20, 86, 50, 11, 26, 121, 18, 38, 4, 31, 199, 34, 12, 139, 28, 63, 62, 61, 67, 60, 125, 59, 16, 37, 58, 47, 108, 23, 21, 41, 13, 112, 71, 194, 56, 25, 30, 302, 19, 166, 46, 55, 150, 83, 116, 219, 73, 90, 140, 164, 54, 207

Offset: 1

M. F. Hasler, Feb 01 2009

Cf. A155891, A155892, A018799, A076219.

A155893(n)={ local( F=1,k=1 ); while( F\1!=n, F*=k++; while( F>=n+1, F/=8 )); k}

A155895 Least positive integer such that a(n)! starts with n, both written in base 5.

Original entry on oeis.org

1, 2, 17, 4, 6, 3, 29, 7, 10, 19, 31, 8, 18, 14, 37, 61, 81, 17, 33, 11, 63, 94, 9, 4, 13, 24, 16, 6, 27, 22, 52, 152, 28, 21, 143, 45, 35, 29, 67, 7, 62, 15, 247, 181, 193, 10, 105, 48, 12, 97, 19, 167, 36, 72, 146, 75, 31, 391, 69, 40, 186, 486, 133, 8, 57, 46, 18, 103, 115

Offset: 1

M. F. Hasler, Feb 01 2009

			a(3) = 17 since 17! is the least factorial to start with digit "3" when written in base 5. a(5) = 6 since 5 = 10[5] (i.e. written in base 5) and 6! = 720 = 625 + 75 + 20 = 10340[5] is the smallest factorial to start with digits "10" when written in base 5.

Cf. A155891--A155894, A018799, A076219.

A155895(n)={ local( F=1,k=1 ); while( F\1!=n, F*=k++; while( F>=n+1, F/=5)); k}

A155896 Least positive integer such that a(n)! starts with n, both written in base 6.

Original entry on oeis.org

1, 2, 5, 4, 8, 3, 9, 26, 52, 18, 25, 10, 33, 53, 32, 15, 13, 20, 30, 5, 17, 42, 7, 4, 23, 43, 78, 50, 28, 113, 8, 44, 19, 334, 59, 100, 195, 27, 22, 14, 91, 172, 371, 16, 93, 9, 12, 102, 66, 26, 46, 95, 279, 303, 69, 331, 272, 52, 286, 79, 264, 298, 18, 21, 190, 104, 47, 138, 255

Offset: 1

M. F. Hasler, Feb 01 2009

			a(3) = 5 since 5!=120 is the least factorial to start with digit "3" when written in base 6. a(8) = 26 since 8 = 12[6] (i.e. written in base 6) and 26! is the smallest factorial to start with digits "12" when written in base 6.

Cf. A155891--A155894, A018799, A076219.

A155896(n)={ local( F=1,k=1 ); while( F\1!=n, F*=k++; while( F>=n+1, F/=6)); k}

cp's OEIS Frontend

A018799 Smallest nonnegative integer m such that m! begins with n in base 10.

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A155891 Least positive integer such that a(n)! starts with n, both written in binary.

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A155892 Least positive integer such that a(n)! starts with n, both written in base 3.

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A155894 Least positive integer such that a(n)! starts with n, both written in base 16.

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A155893 Least positive integer such that a(n)! starts with n, both written in base 8.

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A155895 Least positive integer such that a(n)! starts with n, both written in base 5.

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A155896 Least positive integer such that a(n)! starts with n, both written in base 6.

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