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 A362567 #16 Jun 18 2025 00:58:26
%S A362567 0,3,21,99,375,1137,3267,8595,21891,52965,120087,267843,572145,
%T A362567 1194483,2476743,5037825,9980691
%N A362567 Number of rational solutions to the S-unit equation x + y = 1, where S = {prime(i): 1 <= i <= n}.
%C A362567 Let S = {p_1, p_2, ..., p_n} be a finite set of prime numbers. A rational S-unit is a rational number x such that abs(x) = p_1^k_1 * p_2^k_2 * ... * p_n^k_n for some integers k_1, k_2, ..., k_n.
%C A362567 Thus a(n) is the number of ordered pairs (x,y) of rational numbers such that x+y=1 and v_p(x) = v_p(y) = 0 for all primes p greater than prime(n), i.e., the primes dividing the numerator or denominator of x or y are some subset of the first n prime numbers.
%C A362567 Mahler (1933) first proved that a(n) is finite for all n, with effective bounds first given by Györy (1979).
%H A362567 A. Alvarado, A. Koutsianas, B. Malmskog, C. Rasmussen, C. Vincent, and M. West, <a href="https://arxiv.org/abs/1903.00977">A robust implementation for solving the S-unit equation and several applications</a>, arXiv:1903.00977 [math.NT], 2019.
%H A362567 B. M. M. de Weger, <a href="https://doi.org/10.1016/0022-314X(87)90088-6">Solving exponential Diophantine equations using lattice basis reduction algorithms</a>, J. Number Theory 26 (1987), no. 3, 325-367.
%H A362567 R. von Känel and B. Matschke, <a href="https://arxiv.org/abs/1605.06079">Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture</a>, arXiv:1605.06079 [math.NT], 2016.
%F A362567 a(n) = 6*A362593(n) - 3 if n > 0.
%e A362567 For n = 1, the a(1) = 3 solutions are -1 + 2 = 1, 1/2 + 1/2 = 1, and 2 + -1 = 1.
%e A362567 For n = 2, the a(2) = 21 solutions are -8 + 9 = 1, -3 + 4 = 1, -2 + 3 = 1, -1 + 2 = 1, -1/2 + 3/2 = 1, -1/3 + 4/3 = 1, -1/8 + 9/8 = 1, 1/9 + 8/9 = 1, 1/4 + 3/4 = 1, 1/3 + 2/3 = 1, 1/2 + 1/2 = 1, 2/3 + 1/3 = 1, 3/4 + 1/4 = 1, 8/9 + 1/9 = 1, 9/8 + -1/8 = 1, 4/3 + -1/3 = 1, 3/2 + -1/2 = 1, 2 + -1 = 1, 3 + -2 = 1, 4 + -3 = 1, and 9 + -8 = 1.
%o A362567 (SageMath)
%o A362567 from sage.rings.number_field.S_unit_solver import solve_S_unit_equation
%o A362567 def a(n):
%o A362567 Q = CyclotomicField(1)
%o A362567 S = Q.primes_above(prod([p for p in Primes()[:n]]))
%o A362567 sols = len(solve_S_unit_equation(Q, S))
%o A362567 return 2*sols - 1
%Y A362567 Cf. A002071, A361661, A362593.
%K A362567 nonn,more
%O A362567 0,2
%A A362567 _Robin Visser_, Apr 25 2023