A052428 -id:A052428 - cp's OEIS frontend

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.

1, 11, 24, 30, 31, 32, 37, 38, 43, 52, 53, 54, 55, 59, 60, 61, 65, 73, 75, 80, 91

Offset: 1

Jud McCranie, Dec 16 1999

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)

			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.

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

A subsequence of A052428.

Cf. A051882.

Name clarified by Juhani Heino, Jul 14 2017

A297895 Numbers that can be partitioned into squares of distinct integers whose reciprocals sum to 1.

Original entry on oeis.org

1, 49, 200, 338, 418, 445, 486, 489, 530, 569, 609, 610, 653, 770, 775, 804, 845, 855, 898, 899, 939, 978, 1005, 1019, 1049, 1065, 1085, 1090, 1134, 1194, 1207, 1213, 1214, 1254, 1281, 1308, 1356, 1374, 1379, 1382, 1415, 1434, 1442, 1457, 1458, 1459, 1475, 1499, 1502, 1522, 1543, 1566, 1570, 1582

Offset: 1

Max Alekseyev, Jan 08 2018

nonn

All integers > 8542 = A297896(1) belong to this sequence.

			49 = 2^2 + 3^2 + 6^2, where 1/2 + 1/3 + 1/6 = 1;
200 = 2^2 + 4^2 + 6^2 + 12^2, where 1/2 + 1/4 + 1/6 + 1/12 = 1;
338 = 2^2 + 3^2 + 10^2 + 15^2, where 1/2 + 1/3 + 1/10 + 1/15 = 1.

Max Alekseyev, Table of n, a(n) for n = 1..5000
Max Alekseyev (2019). On partitions into squares of distinct integers whose reciprocals sum to 1. In: J. Beineke, J. Rosenhouse (eds.) The Mathematics of Various Entertaining Subjects: Research in Recreational Math, Volume 3, Princeton University Press, pp. 213-221. ISBN 978-0-691-18257-5 DOI:10.2307/j.ctvd58spj.18 Preprint arXiv:1801.05928 [math.NT], 2018.
Max Alekseyev, List of all representable numbers in the interval [1,54533] and their representations (see Alekseyev 2019 paper for details)

Cf. A051882, A051909, A052428, A297896, A303400.

For n >= 4496, a(n) = n + 4047.

A303400 Numbers that can be partitioned into squares of distinct integers >= 6, whose reciprocals sum to 1.

Original entry on oeis.org

2579, 3633, 3735, 3868, 3948, 4237, 4469, 4544, 4588, 4663, 4678, 4789, 4840, 4913, 4928, 4959, 4995, 5024, 5094, 5104, 5180, 5344, 5393, 5584, 5625, 5642, 5689, 5704, 5717, 5744, 5790, 5799, 5804, 5808, 5856, 5865, 5877, 5900, 5909, 5921, 5923, 5938, 5952, 5953, 5957, 5967, 5984, 6013, 6032, 6034, 6040, 6049, 6114, 6130, 6148, 6150, 6196, 6200, 6234, 6246, 6248, 6272, 6284, 6287

Offset: 1

Max Alekseyev, Apr 23 2018

Also, 6-representable numbers (Alekseyev 2019).

All integers > 15707 = A297896(6) belong to this sequence.

			2579 = 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 12^2 + 14^2 + 15^2 + 18^2 + 24^2 + 28^2, where 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 + 1/18 + 1/24 + 1/28 = 1.

Max Alekseyev, Table of n, a(n) for n = 1..10000
Max Alekseyev (2019). On partitions into squares of distinct integers whose reciprocals sum to 1. In: J. Beineke, J. Rosenhouse (eds.) The Mathematics of Various Entertaining Subjects: Research in Recreational Math, Volume 3, Princeton University Press, pp. 213-221. ISBN 978-0-691-18257-5 DOI:10.2307/j.ctvd58spj.18 Preprint arXiv:1801.05928 [math.NT], 2018.
Max Alekseyev, List of all 6-representable numbers in the interval [1,76744] and their 6-representations (see Alekseyev 2019 paper for details)

Cf. A051882, A051909, A052428, A297895, A297896.

For n >= 5484, a(n) = n + 10224.

cp's OEIS Frontend

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.

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A297895 Numbers that can be partitioned into squares of distinct integers whose reciprocals sum to 1.

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A303400 Numbers that can be partitioned into squares of distinct integers >= 6, whose reciprocals sum to 1.

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