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A254791 Nontrivial solutions to n = sigma(a) = sigma(b) (A000203) and rad(a) = rad(b) (A007947) with a != b.

%I A254791 #15 Feb 24 2022 11:28:43
%S A254791 4800,142800,1909440,32948784,210313800,993938400,1069286400,
%T A254791 1264808160,1309463064,2281635216,3055104000,3250790400
%N A254791 Nontrivial solutions to n = sigma(a) = sigma(b) (A000203) and rad(a) = rad(b) (A007947) with a != b.
%C A254791 On the term "nontrivial":
%C A254791 If a !=b, sigma(a) = sigma(b) and rad(a) = rad(b) then sigma(a*x) = sigma(b*x) and rad(n*x) = rad(m*x) when gcd(a, b) = gcd(a,x) = gcd(b,x) = 1. So each general solution to the stated problem could generate an infinitude of constructed, "trivial" solutions. So we will limit ourselves to the more interesting "nontrivial" solutions. Precisely, if rad(a) = rad(b) = Product(p(i)), we can write a = Product(p(i)^a(i)), b = Product(p(i)^b(i)) and in this context, a(i) != b(i) for each i in order to have a nontrivial solution.
%C A254791 There is another type of trivial solution, if n can be expressed as the product of two or more smaller solutions, it would be considered a composite solution but still trivial.
%C A254791 The smallest composite solution is below:
%C A254791 210313800: 131576362 = 2 * 17 * 157^3 and 98731648 = 2^7 * 17^3 * 1573250790400: 2196937295 = 5 * 7^3 * 31^3 * 43 and 2156627375 = 5^3 * 7 * 31 * 43^3. Note: the common rads for the two pairs have no factors in common so we have these "trivial" composite solutions below.
%C A254791 sigma(131576362 * 2196937295) = sigma(98731648 * 2156627375) = sigma(131576362 * 2156627375) = sigma(98731648 * 2196937295) = 683686082027520000.
%e A254791 Sigma => Pair of distinct integers 4800 => 2058 = 2 * 3 * 7^3 and 1512 = 2^3 * 3^3 * 7142800 => 52728 = 2^3 * 3 * 13^3 and 44928 = 2^7 * 3^3 * 131909440 => 1038230 = 2 * 5 * 47^3 and 752000 = 2^7 * 5^3 * 4732948784 => 10825650 = 2 * 3^9 * 5^2 * 11 and 8624880 = 2^4 * 3^4 * 5 * 11^3210313800 => 131576362 = 2 * 17 * 157^3 and 98731648 = 2^7 * 17^3 * 157993938400 => 336110688 = 2^5 * 3^3 * 73^3 and 326965248 = 2^11 * 3^7 * 73.
%e A254791 The pairs that contribute to the solution each have the same rad or squarefree kernel and they are "nontrivial" because within a pair for the same prime, none of the exponents match.
%Y A254791 Cf. A000203, A007947.
%Y A254791 Subsequence of A254035. Cf. also A255334, A255425, A255426.
%K A254791 nonn
%O A254791 1,1
%A A254791 _Fred Schneider_, Feb 07 2015