cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Teena Carroll

Teena Carroll's wiki page.

Teena Carroll has authored 2 sequences.

A192881 Number of terms for the shortest Egyptian fraction representation of 1 starting with 1/n.

Original entry on oeis.org

1, 3, 5, 8, 10, 11, 13, 15, 17, 19
Offset: 1

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Author

Teena Carroll, Jul 11 2011

Keywords

Comments

An Egyptian fraction representation of a rational number a/b is a list of distinct unit fractions with sum a/b.

Examples

			Since 1/3 + 1/4 + 1/5 + 1/6 + 1/20 = 1, we see that a(3) <= 5.  We know the maximum sum of 4 distinct unit fractions (1/3 or less) is 19/20, so this shows a(3)=5. An Egyptian fraction decomposition of 1 starting with 1/4 must have at least 8 terms; however, the expressions need not be unique, as all three of 1 = 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/230 + 1/57960, 1 = 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/231 + 1/27720 and 1 = 1/4 + 1/5 + 1/6 + 1/9 + 1/10 + 1/15 + 1/18 + 1/20 achieve this bound. - _Teena Carroll_, _Haoqi Chen_ and _Javier Múgica_

Links

Crossrefs

Cf. A103762, A294651.

Formula

a(n) >= A103762(n) - n + 1.

Extensions

Two more terms from Javier Múgica, Dec 18 2017

A118017 Largest denominators in even Egyptian fraction representation of 1.

Original entry on oeis.org

12, 84, 3612, 6526884, 21300113901612
Offset: 1

Views

Author

Teena Carroll, Jul 06 2011

Keywords

Comments

An Egyptian Fraction representation of a rational number a/b is a list of distinct unit fractions with sum a/b. We will call it an even Egyptian Fraction representation if only even integers are used as denominators. The n-th term of this sequence gives the largest denominators that appear in an Egyptian Fraction Representation of one with length n+3. For instance, 1/2 + 1/4 + 1/6 + 1/12 gives a 4-term even representation of one, which is the shortest possible Egyptian even fraction representation of one.
This sequence can be derived from the Sylvester sequence (A000058). If s(n) represents the Sylvester sequence, s(n)-1 is the largest denominator appearing in an n term Egyptian fraction representation of 1. There is a one-to-one correspondence between k-term representations and (k+1)-term even representation for k<12. An even representation has to have at least 4 terms, thus a(1) is related to s(3). a(1) = 2*(s(3) - 1), etc.
Excluding the first two terms of the Pythagorean spiral sequence (A053631) yields this sequence.

Examples

			a(2) = 1/2*12^2 + 12 so a(2) = 84.

References

  • 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

Crossrefs

Cf. A000058, A007018, A053631.
A053631 and A127690 are very similar to this sequence.

Formula

a(n+1) = 1/2*a(n)^2 + a(n).

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