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

4800, 142800, 1909440, 32948784, 210313800, 993938400, 1069286400, 1264808160, 1309463064, 2281635216, 3055104000, 3250790400

Offset: 1

Author

Fred Schneider, Feb 07 2015

Keywords

• nonn

Comments

On the term "nontrivial":

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.

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.

The smallest composite solution is below:

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.

sigma(131576362 * 2196937295) = sigma(98731648 * 2156627375) = sigma(131576362 * 2156627375) = sigma(98731648 * 2196937295) = 683686082027520000.

Examples

			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.
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.
		

Crossrefs

Cf. A000203, A007947.

Subsequence of A254035. Cf. also A255334, A255425, A255426.