A159991 -id:A159991 - cp's OEIS frontend

A100403 Digital root of 6^n.

1, 6, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Cino Hilliard, Dec 31 2004

Also the digital root of k^n for any k == 6 (mod 9). - Timothy L. Tiffin, Dec 02 2023

			For n=8, the digits of 6^8 = 1679616 sum to 36, whose digits sum to 9. So, a(8) = 9. - _Timothy L. Tiffin_, Dec 01 2023

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000400, A001024, A007953, A009968, A009977, A009986, A010888, A100401, A159991.

PadRight[{1, 6}, 100, 9] (* Timothy L. Tiffin, Dec 03 2023 *)

a(n) = if( n<2, [1,6][n+1], 9); \\ Joerg Arndt, Dec 03 2023

From Timothy L. Tiffin, Dec 01 2023: (Start)
a(n) = 9 for n >= 2.
G.f.: (1+5x+3x^2)/(1-x).
a(n) = A100401(n) for n <> 1.
a(n) = A010888(A000400(n)) = A010888(A001024(n)) = A010888(A009968(n)) = A010888(A009977(n)) = A010888(A009986(n)) = A010888(A159991(n)). (End)
E.g.f.: 9*exp(x) - 3*x - 8. - Elmo R. Oliveira, Aug 09 2024
a(n) = A007953(6*a(n-1)) = A010888(6*a(n-1)). - Stefano Spezia, Mar 20 2025

A070197 Base-60 (or sexagesimal or Babylonian) expansion of sqrt(2).

Original entry on oeis.org

1, 24, 51, 10, 7, 46, 6, 4, 44, 50, 28, 51, 20, 34, 26, 20, 4, 31, 2, 38, 30, 53, 27, 38, 34, 5, 46, 18, 24, 29, 40, 16, 7, 16, 8, 56, 52, 55, 33, 23, 4, 47, 56, 56, 45, 38, 49, 18, 57, 50, 33, 52, 37, 54, 34, 48, 24, 21, 7, 17, 14, 49, 19, 7, 29, 21, 53, 42, 42, 11, 57, 11, 11

Offset: 1

Eric W. Weisstein, May 01 2002

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 182.
David Flannery, The Square Root of 2, A Dialogue Concerning A Number And A Sequence, Copernicus Books, NY, 2006, pp. 32-33.

G. C. Greubel, Table of n, a(n) for n = 1..10000
John Baez and Richard Elwes, Babylon and the Square Root of 2, useful comments by Greg Egan et al., Dec 02 2011
Jason Kimberley, Index of expansions of sqrt(d) in base b
Eric Weisstein's World of Mathematics, Pythagoras's Constant
Eric Weisstein's World of Mathematics, Sexagesimal

Cf. A002193.
Cf. A159991. - Reinhard Zumkeller, May 02 2009
Cf. A060707. - Jason Kimberley, Dec 05 2012

Mathematica
```
RealDigits[Sqrt[2], 60, 73][[1]]
```

Entry revised by Robert G. Wilson v, Aug 21 2006

Offset corrected by R. J. Mathar, Feb 05 2009

A091720 Babylonian sexagesimal (base 60) expansion of 1/7.

Original entry on oeis.org

8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17, 8, 34, 17

Offset: 0

Jeppe Stig Nielsen, Feb 01 2004

Index entries for linear recurrences with constant coefficients, signature (0, 0, 1).

Cf. A020806, A091721, A091722, A055643.

Cf. A159991. - Reinhard Zumkeller, May 02 2009

RealDigits[ 1/7, 60, 75] [[1]] (* Robert G. Wilson v, Feb 02 2004 *)

A125628 Version of sexagesimal expansion of 2*Pi given by the Persian mathematician Al-Kashi in the 15th Century.

Original entry on oeis.org

6, 16, 59, 28, 1, 34, 51, 46, 14, 50

Offset: 1

Mohammad K. Azarian, Apr 08 2008

The last digit is rounded up from 49, 55 (cf. A091649). - Georg Fischer, Aug 04 2021

			2*Pi ~= 6; 16, 59, 28, 1, 34, 51, 46, 14, 50.

Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.
Mohammad K. Azarian, The Introduction of Al-Risala al-Muhitiyya: An English Translation, International Journal of Pure and Applied Mathematics, Vol. 57, No. 6, 2009 , pp. 903-914.
Mohammad K. Azarian, Al-Kashi's Fundamental Theorem, International Journal of Pure and Applied Mathematics, Vol. 14, No. 4, 2004, pp. 499-509. Mathematical Reviews, MR2005b:01021 (01A30), February 2005, p. 919. Zentralblatt MATH, Zbl 1059.01005.
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.

Wikipedia, Computation of 2 Pi

Cf. A000796, A011545, A037000, A091649.

Cf. A159991. - Reinhard Zumkeller, May 02 2009

A257895 Square array read by ascending antidiagonals where T(n,k) is the mean number of maxima in a set of n random k-dimensional real vectors (denominators).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 12, 36, 8, 1, 1, 60, 144, 216, 16, 1, 1, 20, 3600, 1728, 1296, 32, 1, 1, 140, 3600, 216000, 20736, 7776, 64, 1, 1, 280, 176400, 72000, 12960000, 248832, 46656, 128, 1, 1, 2520, 705600, 24696000, 12960000, 777600000

Offset: 1

Jean-François Alcover, May 12 2015

			Array of fractions begins:
1,      1,          1,             1,                 1,                    1, ...
1,    3/2,        7/4,          15/8,             31/16,                63/32, ...
1,   11/6,      85/36,       575/216,         3661/1296,           22631/7776, ...
1,  25/12,    415/144,     5845/1728,       76111/20736,        952525/248832, ...
1, 137/60, 12019/3600, 874853/216000, 58067611/12960000, 3673451957/777600000, ...
1,  49/20, 13489/3600,  336581/72000, 68165041/12960000,   483900263/86400000, ...
...
Row 2 (denominators) is A000079 (powers of 2),
Row 3 is A000400 (powers of 6),
Row 4 is A001021 (powers of 12),
Row 5 is A159991,
Row 6 is not in the OEIS.
Column 2 (denominators) is A002805 (denominators of harmonic numbers),
Column 3 is A051418 (lcm(1..n)^2),
Column 4 is not in the OEIS.

Zhi-Dong Bai, Chern-Ching Chao, Hsien-Kuei Hwang and Wen-Qi Liang, On the variance of the number of maxima in random vectors and its applications, The Annals of Applied Probability 1998, Vol. 8, No. 3, 886-895.
O. E. Barndorff-Nielsen and M. Sobel, On the distribution of the number of admissible points in a vector random sample. Theory Probab. Appl. 11 249-269.

Cf. A257894 (numerators).

T[n_, k_] := Sum[(-1)^(j - 1)*j^(1 - k)*Binomial[n, j], {j, 1, n}]; Table[T[n - k + 1, k] // Denominator, {n, 1, 12}, {k, 1, n}] // Flatten

T(n,k) = Sum_{j=1..n} (-1)^(j-1)*j^(1-k)*binomial(n,j).

cp's OEIS Frontend

A100403 Digital root of 6^n.

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A070197 Base-60 (or sexagesimal or Babylonian) expansion of sqrt(2).

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A091720 Babylonian sexagesimal (base 60) expansion of 1/7.

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A125628 Version of sexagesimal expansion of 2*Pi given by the Persian mathematician Al-Kashi in the 15th Century.

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A257895 Square array read by ascending antidiagonals where T(n,k) is the mean number of maxima in a set of n random k-dimensional real vectors (denominators).

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