A169820 -id:A169820 - cp's OEIS frontend

A167209 a(0) = 0; for n>=0, a(n+1) = a(n) + x where x is the smallest nonnegative number which is not equal to a(i) +/- a(k) or a(i)/2 for any 0<=i<=n, 0<=k<=n.

0, 1, 4, 10, 17, 29, 44, 67, 91, 117, 148, 180, 215, 251, 288, 329, 371, 420, 471, 523, 576, 631, 687, 746, 806, 870, 935, 1004, 1074, 1146, 1221, 1297, 1376, 1456, 1538, 1623, 1709, 1802, 1896, 1993, 2092, 2194, 2300, 2409, 2519, 2631, 2753, 2876, 3001, 3127, 3255, 3385, 3518, 3655, 3796, 3939

Offset: 0

N. J. A. Sloane, Sep 03 2014

nonn

A maximal expulsion set of natural numbers.

Note that this definition is different from that of A047699.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
K. S. Brown, Expulsion Sets
Rémy Sigrist, C program for A167209

Cf. A169799, A169820, A047699.

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Initial value changed to a(0)=0 by N. J. A. Sloane, Feb 18 2019 (otherwise the definition did not match the terms).

A169799 First differences of A167209.

Original entry on oeis.org

1, 3, 6, 7, 12, 15, 23, 24, 26, 31, 32, 35, 36, 37, 41, 42, 49, 51, 52, 53, 55, 56, 59, 60, 64, 65, 69, 70, 72, 75, 76, 79, 80, 82, 85, 86, 93, 94, 97, 99, 102, 106, 109, 110, 112, 122, 123, 125, 126, 128, 130, 133, 137, 141, 143, 145, 150, 153, 154, 157, 166, 168, 172, 173, 174, 178, 185, 188, 193, 195, 199, 200

Offset: 1

N. J. A. Sloane, Sep 03 2014

nonn

K. S. Brown, Expulsion Sets

Cf. A167209, A169820.

Definition changed because of new offset in A167209. - N. J. A. Sloane, Feb 18 2019

A130738 Greedy odd Egyptian fraction representation of 1 (without repeats).

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 23, 721, 979007, 661211444787, 622321538786143185105739, 511768271877666618502328764212401495966764795565, 209525411280522638000804396401925664136495425904830384693383280180439963265695525939102230139815

Offset: 1

Jon Wild, Jul 06 2007

a(n) is the largest odd Egyptian fraction as yet unused, such that the sum of the Egyptian fractions so far does not exceed 1. The sum of a(n) is a greedy representation (greedy because each step bites off as much as possible) of 1, using only odd Egyptian fractions, all distinct.

Terms a(11)-a(13) were found by David Eppstein (see posting from Nov 09 1996), who says that he found them by applying EgyptOddGreedy[2/3,5] from his Egyptian fractions notebook.

			E.g. a(8)=721 because 1/721 is the largest odd Egyptian fraction less than 1-1/a(1)-1/a(2)-1/a(3)-1/a(4)-1/a(5)-1/a(6)-1/a(7).
1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/13 + 1/23 + 1/721 + 1/979007 + 1/661211444787 + 1/622321538786143185105739 + 1/511768271877666618502328764212401495966764795565 + 1/209525411280522638000804396401925664136495425904830384693383280180439963265695525939102230139815 = 1.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342
R. K. Guy, Unsolved Problems Number Theory, Sect D11.

David Eppstein, Egyptian fractions
David Eppstein, Egyptian fractions, Discussion, Nov 09 1996.
Index entries for sequences related to Egyptian fractions

Cf. A002966, A169820.

Edited and a(11)-a(13) added by N. J. A. Sloane, May 29 2010, at the suggestion of Jan Szejko.

cp's OEIS Frontend

A167209 a(0) = 0; for n>=0, a(n+1) = a(n) + x where x is the smallest nonnegative number which is not equal to a(i) +/- a(k) or a(i)/2 for any 0<=i<=n, 0<=k<=n.

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A169799 First differences of A167209.

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A130738 Greedy odd Egyptian fraction representation of 1 (without repeats).

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