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.

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

Author

Wouter van Doorn, Feb 05 2025

Keywords

• nonn
• more

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A051882.

Formula

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