This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378270 #31 May 21 2025 10:43:19
%S A378270 1,2,3,7,8,58,59,259,664,3427,3428,73351,73352,298785,7060868,
%T A378270 43070304,43070305,901194373,901194374,32808600352,1204438945226,
%U A378270 2459587779124,2459587779125,96010353352980
%N A378270 Number of partitions of 1 into {1/1^2, 1/2^2, 1/3^2, ..., 1/n^2}.
%C A378270 From _David A. Corneth_, Nov 24 2024: (Start)
%C A378270 Primes n/2 < p <= n occur in exactly one solution namely (p^2) * (1/p^2).
%C A378270 Proof: If the numerator k of k/p^2 is less than p^2 then p divides the denominator of the sum of the Egyptian fractions as p divides no other number <= n. But the goal is have sum 1 i.e. denominator 1 so p cannot be a divisor of the denominator. Consequently this can be reduced to "Number of partitions of 1 into {1/1^2, 1/2^2, ..., 1/(n/2)^2, ..., 1/n^2}" plus number of primes n/2 < p <= n. The denominators for the first part can be cleared turning this into a partitioning problem of positive integers. (End)
%H A378270 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%F A378270 a(p) = a(p-1) + 1 for prime p. - _David A. Corneth_, Nov 22 2024
%e A378270 a(4) = 7 because we have 16 * (1/16) = 12 * (1/16) + 1/4 = 8 * (1/16) + 2 * (1/4) = 4 * (1/16) + 3 * (1/4) = 9 * (1/9) = 4 * (1/4) = 1.
%e A378270 From _David A. Corneth_, Nov 24 2024: (Start)
%e A378270 To find a(12) we can rewrite the problem as "Number of partitions of 1 into {1/1^2, 1/2^2, 1/3^2, 1/4^2, 1/5^2, 1/6^2, 1/8^2, 1/9^2, 1/10^2, 1/12^2} + |{7, 11}|". The lcm of (1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 8^2, 9^2, 10^2, 12^2) is 129600. So this comes a partition problem of (number of partitions of 129600 into parts 129600, 32400, 14400, 8100, 5184, 3600, 2025, 1600, 1296, 900) + |{7, 11}|. (End)
%Y A378270 Cf. A000290, A020473, A038034, A348625, A378271.
%K A378270 nonn,more
%O A378270 1,2
%A A378270 _Ilya Gutkovskiy_, Nov 21 2024
%E A378270 a(12)-a(21) from _David A. Corneth_, Nov 22 2024
%E A378270 a(22)-a(24) from _Jinyuan Wang_, Dec 11 2024