A282622 -id:A282622 - 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

A281863 Alternating powers of 60 and 10 times powers of 60.

Original entry on oeis.org

1, 10, 60, 600, 3600, 36000, 216000, 2160000, 12960000, 129600000, 777600000, 7776000000, 46656000000, 466560000000, 2799360000000, 27993600000000, 167961600000000, 1679616000000000, 10077696000000000, 100776960000000000, 604661760000000000

Offset: 0

Wolfdieter Lang, Feb 19 2017

These numbers are the values for the positions in the Sumerian (and Babylonian) alternating sexagesimal - decimal system (used at least up to 10*60^2 = 36000, but here extended).

For the numbers in this mixed base system see A055643. For the number of symbols needed for representing n see A131650. For the number of digits (including 0) of the representation of n see A282622.

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.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 127.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,60).

Cf. A131650, A055643, A282622.

LinearRecurrence[{0,60},{1,10},21] (* or *) a[0]=1;a[1]=10;a[n_]:=60*a[n-2];Table[a[n],{n,0,20}] (* Indranil Ghosh, Feb 21 2017 *)

Vec((1 + 10*x) / (1 - 60*x^2) + O(x^30)) \\ Colin Barker, Feb 21 2017

a(2*n) = 60^(n/2), a(2*n+1) = 10*60^((n-1)/2), n >= 0.
From Colin Barker, Feb 21 2017: (Start)
a(n) = 60*a(n-2) for n>1.
G.f.: (1 + 10*x) / (1 - 60*x^2). (End)
E.g.f.: cosh(2*sqrt(15)*x) + sqrt(5/3)*sinh(2*sqrt(15)*x). - Stefano Spezia, Sep 08 2024

A131650 Number of symbols in Babylonian numeral representation of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4

Offset: 1

Alain Van Kerckhoven (alain(AT)avk.org), Sep 10 2007

From Wolfdieter Lang, Feb 21 2017: (Start)
For a(1)..a(59) this sequence coincides with A007953.
For the sexagesimal - decimal representation of n see A055643.
The values of the positions is given in A281863.
The sum of the digits of A055643(n) = a(n).
The number of digits of the representation of n is given in A282622. (End)

Wolfdieter Lang, Table of n, a(n) for n = 1..10000
University of St Andrews, Babylonian numerals.
Wikipedia, Babylonian numerals.
Yale Babylonian Collection, High resolution photographs, descriptions and analysis of the root(2) tablet (YBC 7289).

Cf. A006968, A007953, A055643, A281863, A282622.

More terms from Michel Marcus, Jul 12 2013

Terms a(60)-a(81) from Wolfdieter Lang, Feb 21 2017 (to distinguish it from A007953)

cp's OEIS Frontend

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

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A281863 Alternating powers of 60 and 10 times powers of 60.

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Programs

Mathematica

PARI

Formula

A131650 Number of symbols in Babylonian numeral representation of n.

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Author

Keywords

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Crossrefs

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