A100695 -id:A100695 - cp's OEIS frontend

A287636 Indices of records in A100695.

1, 2, 4, 6, 10, 15, 16, 22, 30, 66, 70, 103, 210, 261, 1118, 1625, 2815, 3081, 3088, 6693, 18390, 18762, 32697, 59860, 71355, 155165, 198033, 208790, 262546, 266403, 673708, 697924, 699690, 760750, 1081782, 2993838, 4449916, 4672993, 4692381, 5547049, 6068616

Offset: 1

Amiram Eldar, May 28 2017

nonn

The corresponding denominators are 2, 6, 20, 42, 231, 240, 32640, 34362, 633759288, 1532037276, 4716994695, 100039636784966424, ... The denominator that corresponds to a(45) has 878175 digits.

Amiram Eldar, Table of n, a(n) for n = 1..57

Cf. A100678, A100695, A228136.

a={};dmax=0;n=1;While[Length[a]<40,denom=0; k=n/(n+1); While[ k>0,denom=Ceiling[1/k]; k=k-1/denom]; If[denom>dmax,dmax=denom;a=AppendTo[a,n]];n++];a

A100678 Number of Egyptian fractions in the representation of n/(n+1) via the greedy algorithm.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 3, 4, 5, 3, 4, 4, 4, 4, 5, 3, 4, 4, 4, 4, 5, 4, 6, 4, 4, 5, 5, 4, 5, 5, 5, 4, 5, 3, 4, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 4, 5, 5, 5, 4, 5, 6, 4, 5, 5, 5, 6, 5, 5, 4, 6, 5, 5, 5, 5, 5, 5, 4, 5, 6, 6, 5, 6, 4, 5, 6, 5, 6, 6, 5, 4, 5, 5, 5, 5, 5

Offset: 1

Pahikkala Jussi, Dec 06 2004

nonn

a(n) = length of n-th row in table A247765. - Reinhard Zumkeller, Sep 25 2014

			a(16) = 5 because 16/17 = 1/2 + 1/3 + 1/10 + 1/128 + 1/32640.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..100 from Reinhard Zumkeller)
Wikipedia, Greedy algorithm for Egyptian fractions

Cf. A100695.

Cf. A247765.

a100678 = length . a247765_row -- Reinhard Zumkeller, Sep 25 2014

A100678(n)={ my(x = n/(n+1), nb = 1); while(x -= 1/ceil(1/x), nb++); nb} \\ Michel Marcus, Aug 12 2013, minor edits by M. F. Hasler, Sep 25 2014

More terms from M. F. Hasler, Sep 25 2014

A247765 Table of denominators in the Egyptian fraction representation of n/(n+1) by the greedy algorithm.

Original entry on oeis.org

2, 2, 6, 2, 4, 2, 4, 20, 2, 3, 2, 3, 42, 2, 3, 24, 2, 3, 18, 2, 3, 15, 2, 3, 14, 231, 2, 3, 12, 2, 3, 12, 156, 2, 3, 11, 231, 2, 3, 10, 2, 3, 10, 240, 2, 3, 10, 128, 32640, 2, 3, 9, 2, 3, 9, 342, 2, 3, 9, 180, 2, 3, 9, 126, 2, 3, 9, 99, 2, 3, 9, 83, 34362

Offset: 1

Reinhard Zumkeller, Sep 25 2014

A100678(n) = length of n-th row;

T(n, A100678(n)) = A100695(n).

			.   1:  2
.   2:  2, 6
.   3:  2, 4
.   4:  2, 4, 20
.   5:  2, 3
.   6:  2, 3, 42
.   7:  2, 3, 24
.   8:  2, 3, 18
.   9:  2, 3, 15
.  10:  2, 3, 14, 231
.  11:  2, 3, 12
.  12:  2, 3, 12, 156
.  13:  2, 3, 11, 231
.  14:  2, 3, 10
.  15:  2, 3, 10, 240
.  16:  2, 3, 10, 128, 32640
.  17:  2, 3,  9
.  18:  2, 3,  9, 342
.  19:  2, 3,  9, 180
.  20:  2, 3,  9, 126

Seiichi Manyama, Rows n = 1..1000 of triangle, flattened (Rows n = 1..100 from Reinhard Zumkeller)

Cf. A100678, A100695.

import Data.Ratio ((%), numerator, denominator)
a247765 n k = a247765_tabf !! (n-1) !! (k-1)
a247765_tabf = map a247765_row [1..]
a247765_row n = f (map recip [2..]) (n % (n + 1)) where
   f es x | numerator x == 1 = [denominator x]
          | otherwise        = g es
          where g (u:us) | u <= x    = (denominator u) : f us (x - u)
                         | otherwise =  g us

A287434 Largest denominator used in the Egyptian fraction representation of 1-1/(2n+1) by the odd greedy expansion algorithm, without repeats.

Original entry on oeis.org

45, 24885, 315, 45, 340725, 196365, 15, 10965, 196365, 10465, 1652115781968795, 340725, 25245, 3976914451825623169001741646052688658398236092769201887156089117865, 15345, 13695, 6232413355673505, 79365

Offset: 1

Amiram Eldar, May 30 2017

nonn

The odd version of A100695.

			For n = 2, 1-1/(2n+1) = 4/5 = 1/3 + 1/5 + 1/7 + 1/9 + 1/79 + 1/24885, thus a(2) = 24885.

Amiram Eldar, Table of n, a(n) for n = 1..269
Kevin Brown, Odd-Greedy Unit Fraction Expansions.
Wikipedia, Odd greedy expansion.

Cf. A100695, A286720.

odd[n_]:=If[OddQ[n],n,n+1];a={};For[n=0,n<100,n++;dlast=0;k=2n/(2n+1);s1=0; While[k>0,s2=odd[Ceiling[1/k]]; If[s2==s1,s2+=2]; dlast=s2; k=k-1/s2; s1=s2];a=AppendTo[a,dlast]];a

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A287636 Indices of records in A100695.

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A100678 Number of Egyptian fractions in the representation of n/(n+1) via the greedy algorithm.

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A247765 Table of denominators in the Egyptian fraction representation of n/(n+1) by the greedy algorithm.

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Haskell

A287434 Largest denominator used in the Egyptian fraction representation of 1-1/(2n+1) by the odd greedy expansion algorithm, without repeats.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica