A051882 -id:A051882 - cp's OEIS frontend

A052428 Strict Egyptian numbers (complement of A051882).

1, 11, 24, 30, 31, 32, 37, 38, 43, 45, 50, 52, 53, 54, 55, 57, 59, 60, 61, 62, 64, 65, 66, 67, 69, 71, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111

Offset: 1

Eric W. Weisstein

Max A. Alekseyev, On partitions into squares of distinct integers whose reciprocals sum to 1, The Mathematics of Various Entertaining Subjects: Volume 3: The Magic of Mathematics, Part III Algebra and Number Theory (2019), 213-221.
R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3:4 (1963), pp. 435-441.
Eric Weisstein's World of Mathematics, Egyptian Number.
Index entries for sequences related to Egyptian fractions
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A051882, A028229.

Note that A051909 is a subset (but is strictly different).

strictEgyptianQ[m_]:=!Length[Select[IntegerPartitions[m, Ceiling[(Sqrt[8 m + 1] - 1) / 2]], Length[#]==Length[Union[#]]&&1==Plus@@(1/#)&, 1]]>0; Reap[Do[If[!strictEgyptianQ[m], Print[m]; Sow[m]], {m, 1,100}]][[2, 1]] (* Vincenzo Librandi, Jul 16 2017 *)

a(n) = n + 47 for n > 30. - Charles R Greathouse IV, Jun 19 2024

A028229 Call m Egyptian if we can partition m = x_1+x_2+...+x_k into positive integers x_i such that Sum_{i=1..k} 1/x_i = 1; sequence gives all non-Egyptian numbers.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 12, 13, 14, 15, 19, 21, 23

Offset: 1

N. J. A. Sloane, Jud McCranie

Graham showed that every number >=78 is strict-sense Egyptian.

			1=1/3+1/3+1/3, so 3+3+3=9 is Egyptian.

J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 147.
See also R. K. Guy, Unsolved Problems Number Theory, Sect. D11.

R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3:4 (1963), pp. 435-441. doi:10.1017/S1446788700039045
Eric Weisstein's World of Mathematics, Egyptian Number.
Index entries for sequences related to Egyptian fractions

Cf. A051882. Complement gives A125726.

egyptianQ[n_] := Select[ IntegerPartitions[n], Total[1/#] == 1 &, 1] =!= {}; A028229 = Reap[ Do[ If[ !egyptianQ[n], Sow[n]], {n, 1, 40}]][[2, 1]] (* Jean-François Alcover, Feb 23 2012 *)

A051907 Number of ways to express 1 as the sum of distinct unit fractions such that the sum of the denominators is n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 0, 4, 1, 3, 4, 0, 2, 0, 6, 0, 1, 2, 1, 3, 0, 4, 2, 1, 5, 5, 3, 2, 3, 3, 5, 5, 5, 2, 1, 12, 5, 4, 11, 4, 5, 2, 11, 3, 5

Offset: 1

Jud McCranie, Dec 16 1999

nonn

			1 = 1/2+1/4+1/9+1/12+1/18 = 1/2+1/5+1/6+1/12+1/20. The sum of the denominators of each of these is 45, these are the only 2 with sum of denominators = 45, so a(45)=2.

Seiichi Manyama, Table of n, a(n) for n = 1..150
Index entries for sequences related to Egyptian fractions

A051882 lists n such that a(n)=0.

Cf. A051908.

R. L. Graham showed that a(n)>0 for n>77.

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.

Original entry on oeis.org

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.

A380791 For a positive rational x, let k(x) be the smallest positive integer such that all k >= k(x) have a partition into distinct parts with reciprocal sum equal to x. The n-th term in this sequence is equal to the number of x with k(x) equal to n.

Original entry on oeis.org

2, 2, 2, 1, 2, 4, 5, 5, 7, 7, 5, 12, 18, 22, 32, 38, 41, 48, 57, 76, 82, 74, 97, 117, 155, 170, 194, 228, 277, 306, 332, 430, 473, 483, 510

Offset: 66

Wouter van Doorn, Feb 05 2025

R. L. Graham proved that every positive integer k >= 78 can be written as a sum a_1 + a_2 + ... + a_r of distinct positive integers, such that 1/a_1 + 1/a_2 + ... + 1/a_r is equal to 1. More generally, he showed that for every positive rational x there exists a k(x) such that all k >= k(x) can be written as a sum a_1 + a_2 + ... + a_r of distinct positive integers, such that 1/a_1 + 1/a_2 + ... + 1/a_r is equal to x.

			a(66) = 2, as there are 2 positive rationals x (namely 4/5 and 11/12) such that 65 cannot be written as a sum of distinct positive integers whose reciprocal sum is equal to x, but every positive integer larger than or equal to 66 can be written in such a way.
As it turns out, for every positive rational x, there exists a positive integer k >= 65 such that k cannot be written as a sum of distinct positive integers with reciprocal sum equal to x. This is why a(n) = 0 for all n <= 65.

R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3:4 (1963), pp. 435-441.
W. van Doorn, Partitions with prescribed sum of reciprocals: computational results, arXiv:2502.01409 [math.NT], 2025.
W. van Doorn, Partitions with prescribed sum of reciprocals: asymptotic bounds, arXiv:2502.02200 [math.NT], 2025.

Cf. A051882.

a(n) = exp[n^(1/2 + o(1))].

cp's OEIS Frontend

A052428 Strict Egyptian numbers (complement of A051882).

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A028229 Call m Egyptian if we can partition m = x_1+x_2+...+x_k into positive integers x_i such that Sum_{i=1..k} 1/x_i = 1; sequence gives all non-Egyptian numbers.

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A051907 Number of ways to express 1 as the sum of distinct unit fractions such that the sum of the denominators is n.

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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.

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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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A380791 For a positive rational x, let k(x) be the smallest positive integer such that all k >= k(x) have a partition into distinct parts with reciprocal sum equal to x. The n-th term in this sequence is equal to the number of x with k(x) equal to n.

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