A254335 -id:A254335 - cp's OEIS frontend

A055643 Babylonian numbers: integers in base 60 with each sexagesimal digit represented by 2 decimal digits, leading zeros omitted.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110

Offset: 0

Henry Bottomley, Jun 06 2000

From Wolfdieter Lang, Jan 16 2018: (Start)
The symbols used for 0..9 in this base 60 notation are 00, 01, ..., 09, but leading zeros are omitted.
For the Sumerian-Babylonian sexagesimal-decimal number system which uses two positions for each base-60 position filled with only one-digit numbers alternating between ranges of 0 to 9 and 0 to 5 see the link below.
(End)
For n < 1440, US and NATO military time designation of n minutes since midnight. - J. Lowell, Dec 29 2020

Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.
Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.
Georges Ifrah, Histoire Universelle des Chiffres, Paris, 1981.
Georges Ifrah, From one to zero, A universal history of numbers, Viking Penguin Inc., 1985.
Georges Ifrah, Universalgeschichte der Zahlen, Campus Verlag, Frankfurt, New York, 2. Auflage, 1987, pp. 210-221.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Wolfdieter Lang, Sumerian-Babylonian sexagesimal-decimal number system.

Cf. A049872, A131650,
Note also that A250073 = a(A000079(n)), A250089 = a(A051037(n)), A254334 = a(A000244(n)), A254335 = a(A000351(n)), A254336 = a(A011557(n)).
See also A281863 (value of the 0,1,2,...,n-th digit of a(n), counted from the right), A282622 (length of a(n), #digits, for n >= 1).

Array[FromDigits@ Apply[Join, PadLeft[#, 2] & /@ IntegerDigits@ IntegerDigits[#, 60]] &, 71, 0] (* Michael De Vlieger, Jan 11 2018 *)

A055643(n)=fromdigits(digits(n,60),100) \\ M. F. Hasler, Jan 09 2018

def a(n): return n if n < 60 else 100*a(n//60) + n%60
print([a(n) for n in range(71)]) # Michael S. Branicky, Oct 22 2022

a(60*n+r) = 100*a(n) + r, 0 <= r <= 59. - Jianing Song, Oct 22 2022

a(69) and a(70) from WG Zeist, Sep 08 2012

A254334 Powers of 3 in base 60, concatenating the decimal values of the sexagesimal digits.

Original entry on oeis.org

1, 3, 9, 27, 121, 403, 1209, 3627, 14921, 52803, 162409, 491227, 2273721, 7225203, 22083609, 106254827, 319172521, 957521603, 2953364809, 12940502427, 42902311321, 132707334003, 402122410009, 2010408030027, 6031224090121, 18093712270403, 54285137211209

Offset: 0

Michael De Vlieger, Jan 28 2015

Each sexagesimal digit appears as a pair of decimal digits as on a digital clock. Any leading zeros are truncated. Thus decimal 81 appears as "121" and not "0121".

			a(6) = 1209, since 3^6 = 729 = 12 * 60^1 + 9, thus 12:09 in clock-like notation, which becomes 1209 when restricted to numeric characters.

Michael De Vlieger, Table of n, a(n) for n = 0..1200

Cf. A000244 (Powers of 3), A055643 (Babylonian numbers).

Cf. Sexagesimal representations: A250073 (Powers of 2), A254335 (Powers of 5), A254336 (Powers of 10).

f[n_] := FromDigits@ StringJoin[If[# < 10, StringJoin["0", ToString[#]], ToString[#]] & /@ IntegerDigits[3^n, 60]]; Table[f@ i, {i, 0, 26}] (* Michael De Vlieger, Jan 28 2015 *)

a(n) = subst(Pol(digits(3^n, 60)), x, 100); \\ Michel Marcus, Feb 22 2015

def digits(n,b=10): # list of digits of n in base b
    x, y = n, []
    while x >= b:
        x, r = divmod(x,b)
        y.append(r)
    y.append(x)
    return list(reversed(y))
A254334_list = [int(''.join([format(x,'02d') for x in digits(3**i, 60)])) for i in range(10**2)]
# Chai Wah Wu, Mar 14 2015

a(n) = A055643(A000244(n)). - Michel Marcus, Mar 02 2015

A254336 Powers of 10 in base 60, concatenating the decimal values of the sexagesimal digits.

Original entry on oeis.org

1, 10, 140, 1640, 24640, 274640, 4374640, 46174640, 742574640, 11709374640, 125136174640, 2083602574640, 21260029374640, 334200456174640, 3543204922574640, 55713281349374640, 593214421816174640, 9552227030242574640, 139134430302709374640, 1632172505043136174640, 24522541050451602574640

Offset: 0

Michael De Vlieger, Jan 28 2015

Each sexagesimal digit appears as a pair of decimal digits as on a digital clock. Any leading zeros are truncated. Thus decimal 100 appears as "140" and not "0140".

			a(3) = 1640, since 10^3 = 1000 = 16 * 60^1 + 40, thus 16:40 in clock-like notation, which becomes 1640 when restricted to numeric characters.

Michael De Vlieger, Table of n, a(n) for n = 0..600

Cf. A011557 (Powers of 10), A055643 (Babylonian numbers).

Cf. Sexagesimal representations: A250073 (Powers of 2), A254334 (Powers of 3), A254335 (Powers of 5).

f[n_] := FromDigits@ StringJoin[If[# < 10, StringJoin["0", ToString[#]],
ToString[#]] & /@ IntegerDigits[10^n, 60]]; Table[f@ i, {i, 0, 20}] (* Michael De Vlieger, Jan 28 2015 *)

a(n) = subst(Pol(digits(10^n, 60)), x, 100); \\ Michel Marcus, Feb 22 2015

a(n) = A055643(A011557(n)). - Michel Marcus, Mar 02 2015

A250089 5-smooth numbers (A051037) written in base 60, concatenating the decimal values of the sexagesimal digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 100, 104, 112, 115, 120, 121, 130, 136, 140, 148, 200, 205, 208, 215, 224, 230, 240, 242, 300, 312, 320, 336, 345, 400, 403, 410, 416, 430, 448, 500, 520, 524, 600

Offset: 1

Michael De Vlieger, Nov 11 2014

Each pair of digits constitutes the decimal value of a single sexagesimal digit, as on a digital clock, eliminating the colon (:). Any leading zeros are truncated. Thus decimal 64 appears as "104" and not "0104".

			a(28) = 112 since A051037(28) = 72. 72 = 1 * 60 + 12, thus sexagesimal 1,12. Concatenating the decimal values of the sexagesimal places gives "112".

D. E. Knuth, Ancient Babylonian Algorithms, Communications of the ACM 15 (1972): 671-677.

Michael De Vlieger, Table of n, a(n) for n = 1..10540 (Hamming numbers <= 60^10)

Cf. A051037, A250073, A254334, A254335, A254336.

a250089[n_Integer] := FromDigits /@ Map[StringJoin, If[# < 10, StringJoin["0", ToString[#]], ToString[#]] & /@ IntegerDigits[#, 60] & /@ Select[Range[n], Last@Map[First, FactorInteger@#] < 7 &], 2]; a250089[360] (* Michael De Vlieger, Nov 11 2014, after Robert G. Wilson v at A051037 *)
With[{n = 360}, Map[FromDigits@ IntegerDigits[#, MixedRadix[ Flatten@ ConstantArray[{6, 10}, {2 Ceiling@ Log[60, n]}]]] &, Union@ Flatten@ Table[2^p1*3^p2*5^p3, {p1, 0, Log[2, n/(1)]}, {p2, 0, Log[3, n/(2^p1)]}, {p3, 0, Log[5, n/(2^p1*3^p2)]}]]] (* Version 10.2, or *)
With[{n = 360}, FromDigits@ StringJoin@ Map[If[# < 10, StringJoin["0", ToString@ #], ToString@ #] &, IntegerDigits[#, 60]] & /@ Union@ Flatten@ Table[2^p1*3^p2*5^p3, {p1, 0, Log[2, n/(1)]}, {p2, 0, Log[3, n/(2^p1)]}, {p3, 0, Log[5, n/(2^p1*3^p2)]}]] (* Michael De Vlieger, Feb 20 2017 *)

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A055643 Babylonian numbers: integers in base 60 with each sexagesimal digit represented by 2 decimal digits, leading zeros omitted.

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A254334 Powers of 3 in base 60, concatenating the decimal values of the sexagesimal digits.

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A254336 Powers of 10 in base 60, concatenating the decimal values of the sexagesimal digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A250089 5-smooth numbers (A051037) written in base 60, concatenating the decimal values of the sexagesimal digits.

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Keywords

Comments

Examples

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Mathematica