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A362593 Number of coprime positive integer S-unit solutions to a + b = c where a <= b < c, and where S = {prime(1), ..., prime(n)}.

%I A362593 #13 Jun 17 2025 22:24:16
%S A362593 0,1,4,17,63,190,545,1433,3649,8828,20015,44641,95358,199081,412791,
%T A362593 839638,1663449
%N A362593 Number of coprime positive integer S-unit solutions to a + b = c where a <= b < c, and where S = {prime(1), ..., prime(n)}.
%C A362593 Let S = {p_1, p_2, ..., p_n} be a finite set of prime numbers. A positive integer S-unit is a positive integer x such that x = p_1^k_1 * p_2^k_2 * ... * p_n^k_n for some nonnegative integers k_1, k_2, ..., k_n.
%C A362593 Thus a(n) is the number of positive integer triples (a,b,c) such that a + b = c, gcd(a,b,c) = 1, a <= b < c and v_p(a) = v_p(b) = v_p(c) = 0 for all primes p greater than prime(n), i.e., the primes dividing a, b or c are some subset of the first n prime numbers.
%C A362593 Mahler (1933) first proved that a(n) is finite for all n, with effective bounds first given by Györy (1979).
%H A362593 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 A362593 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 A362593 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 A362593 a(n) = (A362567(n) + 3)/6 if n > 0.
%e A362593 For n = 2, the a(2) = 4 solutions are 1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 4, and 1 + 8 = 9.
%e A362593 For n = 3, the a(3) = 17 solutions are 1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 4, 1 + 4 = 5, 1 + 5 = 6, 1 + 8 = 9, 1 + 9 = 10, 1 + 15 = 16, 1 + 24 = 25, 1 + 80 = 81, 2 + 3 = 5, 2 + 25 = 27, 3 + 5 = 8, 3 + 125 = 128, 4 + 5 = 9, 5 + 27 = 32, and 9 + 16 = 25.
%o A362593 (SageMath)
%o A362593 from sage.rings.number_field.S_unit_solver import solve_S_unit_equation
%o A362593 def a(n):
%o A362593     Q = CyclotomicField(1)
%o A362593     S = Q.primes_above(prod([p for p in Primes()[:n]]))
%o A362593     sols = len(solve_S_unit_equation(Q, S))
%o A362593     return (sols + 1)/3
%Y A362593 Cf. A002071 (Case a = 1), A361661, A362567.
%K A362593 nonn,more
%O A362593 0,3
%A A362593 _Robin Visser_, Apr 26 2023