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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.

%I A380791 #28 Mar 03 2025 12:46:50
%S A380791 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,
%T A380791 170,194,228,277,306,332,430,473,483,510
%N 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.
%C A380791 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.
%H A380791 R. L. Graham, <a href="https://doi.org/10.1017/S1446788700039045">A theorem on partitions</a>, J. Austral. Math. Soc. 3:4 (1963), pp. 435-441.
%H A380791 W. van Doorn, <a href="https://arxiv.org/abs/2502.01409">Partitions with prescribed sum of reciprocals: computational results</a>, arXiv:2502.01409 [math.NT], 2025.
%H A380791 W. van Doorn, <a href="https://arxiv.org/abs/2502.02200">Partitions with prescribed sum of reciprocals: asymptotic bounds</a>, arXiv:2502.02200 [math.NT], 2025.
%F A380791 a(n) = exp[n^(1/2 + o(1))].
%e A380791 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.
%e A380791 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.
%Y A380791 Cf. A051882.
%K A380791 nonn,more
%O A380791 66,1
%A A380791 _Wouter van Doorn_, Feb 05 2025