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A051909 Subset of strict Egyptian numbers - there is a unique representation of 1 as the sum of distinct unit fractions with the sum of denominators being these numbers. 4
1, 11, 24, 30, 31, 32, 37, 38, 43, 52, 53, 54, 55, 59, 60, 61, 65, 73, 75, 80, 91 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Note the word "unique" in the definition, which makes this different from A052428. - N. J. A. Sloane, Jul 13 2017

From Juhani Heino, Jul 15 2017: (Start)

McCranie conjectured that the sequence is finite. He was right. Numbers 92-269 all have at least two representations without 15 appearing among the reciprocals (list attached). In the same way that Graham did for strict Egyptian numbers, we can transform each representation (a_n) into an even (2, (2a_n)) and an odd (3, 9, 30, 45, (2a_n)). Because there are no 15's in the original representations, the odd transformation has distinct unit fractions. The same is trivially true for the even transformation because there are no 1's either. So if the original sum is s, the new sums are 2s+2 and 2s+87. 2*92+2 = 186, so the even sums can be obtained from transformations after that. 2*92+87 = 271, so the same holds for odd sums from that point. And as the pairs of original representations were distinct, so are the transformed pairs.

If my programming and reasoning were correct, I proved that the sequence is full (no more members after 91), and my program in the raw mode (not ignoring the 15's) corroborates exactly with McCranie's results. Could somebody prove the uniqueness?

(End)

LINKS

Table of n, a(n) for n=1..21.

R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3:4 (1963), pp. 435-441.

Juhani Heino, List of double representations from 92 to 269

Index entries for sequences related to Egyptian fractions

EXAMPLE

1 = 1/2 + 1/3 + 1/6, 2 + 3 + 6 = 11, there is no other representation of 1 as the sum of distinct unit fractions whose numerators sum to 11, so 11 is in the sequence.

CROSSREFS

A subset of A052428. Cf. A051882.

Sequence in context: A229057 A086710 A052428 * A112134 A063307 A213991

Adjacent sequences:  A051906 A051907 A051908 * A051910 A051911 A051912

KEYWORD

full,nonn,fini

AUTHOR

Jud McCranie, Dec 16 1999

EXTENSIONS

Name clarified by Juhani Heino, Jul 14 2017

STATUS

approved

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Last modified September 19 18:47 EDT 2018. Contains 315210 sequences. (Running on oeis4.)