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A200066 Marks found on the Ishango bone in nondecreasing order. 1
3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 11, 13, 17, 19, 19, 21 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The Ishango bone appears to be the oldest known mathematical object that contains an integer sequence with distinct numbers, dated from 18000 to 20000 BC. The oldest known tally stick is the Lebombo bone which is a piece of baboon fibula with 29 notches, dated 35000 BC. Note that A100000 shows only the "middle column" of Ishango bone. This sequence lists all terms of the three columns of the bone in nondecreasing order. The sequence has 16 terms: ten terms are prime numbers A000040 and six terms are composite numbers A002808. The ratio between the number of primes and the number of composites is equal to 5/8. The sequence contains the first seven odd primes (see A065091). The repeated numbers 5, 11 and 19 are the first three primes of the form n^2 - n - 1 (see A002327), also the first three primes with a Fibonacci primitive root (see A003147). The sum of all terms coincides with the number of primes <= 1000, this is A000720(1000) = 168.

REFERENCES

J. Bogoshi, K. Naidoo and J. Webb, The oldest mathematical artifact, Math. Gazette, 71:458 (1987) 294.

M. Grousson, "Depuis quand compte-t-on ?" in 'Science & Vie', pp. 58-61, No. 1080 2007 Mondadori/Excelsior Publications Paris.

D. Huylebrouck, "L'Afrique, berceau des mathematiques", in Mathematiques exotiques pp. 46-50, Dossier No. 47, Pour La Science 2005 Paris.

D. Huylebrouck and V. Pletser, The Ishango artifact: the missing base 12 link, Proc. Katachi Univ. Symmetry Congress (KUS2), Paper C11, Tsukuba Univ., Japan, 18 Nov. 1999; Forma 14-4, 339-346.

G. G. Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, Penguin Books, London, 1992.

Richard Mankiewicz, The Story of Mathematics, Weidenfeld Nicolson Illustrated; New Ed edition (September 13, 2001).

D. Olivastro, Ancient Puzzles, Chap. 1 "The First Etches" pp. 7-30 Bantam Books NY 1993.

Claudia Zaslavsky, Africa Counts, Lawrence Hill Books, New York, 1973.

LINKS

Table of n, a(n) for n=1..16.

AfricaMaat, Africa: The true cradle of mathematical sciences

Brussels Museum for Natural Sciences, The Ishango Bone Exhibition.

C. K. Caldwell, The Prime Glossary, Ishango bone

Wikipedia, Ishango bone

S. W. Williams, Mathematicians of the Africa diaspora. Ishango bone

EXAMPLE

The three columns (or rows) of the Ishango bone are

[3, 6, 4, 8, 10, 5, 5, 7], [11, 13, 17, 19], [11, 21, 19, 9]. Or [11, 13, 17, 19], [11, 21, 19, 9], [3, 6, 4, 8, 10, 5, 5, 7]. Or [11, 21, 19, 9], [3, 6, 4, 8, 10, 5, 5, 7], [11, 13, 17, 19].

In reverse order, the three columns (or rows) are

[9, 19, 21, 11], [19, 17, 13, 11], [7, 5, 5, 10, 8, 4, 6, 3]. Or [19, 17, 13, 11], [7, 5, 5, 10, 8, 4, 6, 3], [9, 19, 21, 11]. Or [7, 5, 5, 10, 8, 4, 6, 3], [9, 19, 21, 11], [19, 17, 13, 11].

CROSSREFS

Cf. A000040, A000720, A002327, A002808, A003147, A065901, A100000.

Sequence in context: A106501 A065305 A009070 * A227789 A116183 A177744

Adjacent sequences:  A200063 A200064 A200065 * A200067 A200068 A200069

KEYWORD

nonn,fini,full,nice

AUTHOR

Omar E. Pol, Nov 14 2011

STATUS

approved

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Last modified August 21 05:59 EDT 2017. Contains 290862 sequences.