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A337670 Numbers that can be expressed as both Sum x^y and Sum y^x where the x^y are not equal to y^x for any (x,y) pair and all (x,y) pairs are distinct.

%I A337670 #35 Apr 23 2022 16:31:14
%S A337670 432,592,1017,1040,1150,1358,1388,1418,1422,1464,1554,1612,1632,1713,
%T A337670 1763,1873,1889,1966,1968,1973,1990,2091,2114,2190,2291,2320,2364,
%U A337670 2451,2589,2591,2612,2689,2697,2719,2753,2775,2803,2813,2883,3087,3127,3141,3146
%N A337670 Numbers that can be expressed as both Sum x^y and Sum y^x where the x^y are not equal to y^x for any (x,y) pair and all (x,y) pairs are distinct.
%C A337670 Numbers m of form m = Sum_{i=1...k} b_i^e_i = Sum_{i=1...k} e_i^b_i such that b_i^e_i != e_i^b_i, b_i > 1, e_i > 1, k = |{{b_i, e_i}, i = 1, 2, ...}|, k > 1.
%C A337670 Terms of the sequence relate to the Diophantine equation Sum_{i=1...k} x_i = 0, k > 1, x_i != 0, where x_i = (b_i^e_i - e_i^b_i) such that b_i > 1, e_i > 1 and (i != j) => ({b_i, e_i} != {b_j, e_j}). That is, we are observing linear combinations of elements from {(r^n - n^r) : n,r > 1} \ {0}, under given conditions.
%C A337670 For sums with k = 20 terms, one infinite family of examples is known: "2^(2t) + t^(4) + 2^(2t+8) + (t+4)^(4) + 2^(2t+16) + (t+8)^(4) + 2^(2t+32) + (t+16)^(4) + 2^(2t+34) + (t+17)^(4) + 4^(t+1) + (2t+2)^(2) + 4^(t+2) + (2t+4)^(2) + 4^(t+10) + (2t+20)^(2) + 4^(t+14) + (2t+28)^(2) + 4^(t+18) + (2t+36)^(2)" is a term of the sequence, for every t > 4.
%H A337670 Math StackExchange, <a href="https://math.stackexchange.com/q/3795656/318073">Base-Exponent Invariants</a>, 2020.
%H A337670 Matej Veselovac, <a href="/A337670/a337670.txt">PYTHON program for A337670</a>
%H A337670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>.
%e A337670 17 = 2^3 + 3^2 = 3^2 + 2^3 is not in the sequence because {2,3} = {3,2} are not distinct.
%e A337670 25 = 3^3 + 2^4 = 3^3 + 4^2 is not in the sequence because 3^3 = 3^3 and 2^4 = 4^2 are commutative.
%e A337670 The smallest term of the sequence is:
%e A337670   a(1) = 432 = 3^2 + 5^2 + 2^6 + 3^4 + 5^3 + 2^7
%e A337670              = 2^3 + 2^5 + 6^2 + 4^3 + 3^5 + 7^2.
%e A337670 The smallest term that has more than one representation is:
%e A337670   a(11) = 1554 = 3^2 + 7^2 + 6^3 + 2^8 + 4^5
%e A337670                = 2^3 + 2^7 + 3^6 + 8^2 + 5^4,
%e A337670   a(11) = 1554 = 3^2 + 5^2 + 2^6 + 10^2 + 2^7 + 3^5 + 2^8 + 3^6
%e A337670                = 2^3 + 2^5 + 6^2 + 2^10 + 7^2 + 5^3 + 8^2 + 6^3.
%e A337670 Smallest terms with k = 5, 6, 7, 8, 9, 10 summands are:
%e A337670   a(9)  = 1422 = 5^2 + 7^2 + 9^2 + 3^5 + 4^5
%e A337670                = 2^5 + 2^7 + 2^9 + 5^3 + 5^4,
%e A337670   a(1)  = 432  = 3^2 + 5^2 + 2^6 + 3^4 + 5^3 + 2^7
%e A337670                = 2^3 + 2^5 + 6^2 + 4^3 + 3^5 + 7^2,
%e A337670   a(2)  = 592  = 3^2 + 5^2 + 7^2 + 4^3 + 2^6 + 5^3 + 2^8
%e A337670                = 2^3 + 2^5 + 2^7 + 3^4 + 6^2 + 3^5 + 8^2,
%e A337670   a(11) = 1554 = 3^2 + 5^2 + 2^6 + 10^2 + 2^7 + 3^5 + 2^8 + 3^6
%e A337670                = 2^3 + 2^5 + 6^2 + 2^10 + 7^2 + 5^3 + 8^2 + 6^3,
%e A337670   a(14) = 1713 = 3^2 + 2^5 + 6^2 + 8^2 + 4^3 + 2^7 + 3^5 + 2^9 + 5^4
%e A337670                = 2^3 + 5^2 + 2^6 + 2^8 + 3^4 + 7^2 + 5^3 + 9^2 + 4^5,
%e A337670   a(28) = 2451 = 3^2 + 5^2 + 6^2 + 8^2 + 3^4 + 2^7 + 6^3 + 3^5 + 5^4 + 2^10
%e A337670                = 2^3 + 2^5 + 2^6 + 2^8 + 4^3 + 7^2 + 3^6 + 5^3 + 4^5 + 10^2.
%Y A337670 Cf. A337671 (subsequence for k <= 5).
%Y A337670 Cf. A005188 (perfect digital invariants).
%Y A337670 Cf. Perfect powers: A001597, A072103.
%Y A337670 Cf. Commutative powers: A271936.
%Y A337670 Cf. Nonnegative numbers of the form (r^n - n^r), for n,r > 1: A045575.
%Y A337670 Cf. Numbers of the form (r^n - n^r): A024012 (r = 2), A024026 (r = 3), A024040 (r = 4), A024054 (r = 5), A024068 (r = 6), A024082 (r = 7), A024096 (r = 8), A024110 (r = 9), A024124 (r = 10), A024138 (r = 11), A024152 (r = 12).
%K A337670 nonn
%O A337670 1,1
%A A337670 _Matej Veselovac_, Sep 15 2020