A337670 - OEIS

%I #35 Apr 23 2022 16:31:14

%S 432,592,1017,1040,1150,1358,1388,1418,1422,1464,1554,1612,1632,1713,

%T 1763,1873,1889,1966,1968,1973,1990,2091,2114,2190,2291,2320,2364,

%U 2451,2589,2591,2612,2689,2697,2719,2753,2775,2803,2813,2883,3087,3127,3141,3146

%N 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 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 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 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 Math StackExchange, <a href="https://math.stackexchange.com/q/3795656/318073">Base-Exponent Invariants</a>, 2020.

%H Matej Veselovac, <a href="/A337670/a337670.txt">PYTHON program for A337670</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>.

%e 17 = 2^3 + 3^2 = 3^2 + 2^3 is not in the sequence because {2,3} = {3,2} are not distinct.

%e 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 The smallest term of the sequence is:

%e   a(1) = 432 = 3^2 + 5^2 + 2^6 + 3^4 + 5^3 + 2^7

%e              = 2^3 + 2^5 + 6^2 + 4^3 + 3^5 + 7^2.

%e The smallest term that has more than one representation is:

%e   a(11) = 1554 = 3^2 + 7^2 + 6^3 + 2^8 + 4^5

%e                = 2^3 + 2^7 + 3^6 + 8^2 + 5^4,

%e   a(11) = 1554 = 3^2 + 5^2 + 2^6 + 10^2 + 2^7 + 3^5 + 2^8 + 3^6

%e                = 2^3 + 2^5 + 6^2 + 2^10 + 7^2 + 5^3 + 8^2 + 6^3.

%e Smallest terms with k = 5, 6, 7, 8, 9, 10 summands are:

%e   a(9)  = 1422 = 5^2 + 7^2 + 9^2 + 3^5 + 4^5

%e                = 2^5 + 2^7 + 2^9 + 5^3 + 5^4,

%e   a(1)  = 432  = 3^2 + 5^2 + 2^6 + 3^4 + 5^3 + 2^7

%e                = 2^3 + 2^5 + 6^2 + 4^3 + 3^5 + 7^2,

%e   a(2)  = 592  = 3^2 + 5^2 + 7^2 + 4^3 + 2^6 + 5^3 + 2^8

%e                = 2^3 + 2^5 + 2^7 + 3^4 + 6^2 + 3^5 + 8^2,

%e   a(11) = 1554 = 3^2 + 5^2 + 2^6 + 10^2 + 2^7 + 3^5 + 2^8 + 3^6

%e                = 2^3 + 2^5 + 6^2 + 2^10 + 7^2 + 5^3 + 8^2 + 6^3,

%e   a(14) = 1713 = 3^2 + 2^5 + 6^2 + 8^2 + 4^3 + 2^7 + 3^5 + 2^9 + 5^4

%e                = 2^3 + 5^2 + 2^6 + 2^8 + 3^4 + 7^2 + 5^3 + 9^2 + 4^5,

%e   a(28) = 2451 = 3^2 + 5^2 + 6^2 + 8^2 + 3^4 + 2^7 + 6^3 + 3^5 + 5^4 + 2^10

%e                = 2^3 + 2^5 + 2^6 + 2^8 + 4^3 + 7^2 + 3^6 + 5^3 + 4^5 + 10^2.

%Y Cf. A337671 (subsequence for k <= 5).

%Y Cf. A005188 (perfect digital invariants).

%Y Cf. Perfect powers: A001597, A072103.

%Y Cf. Commutative powers: A271936.

%Y Cf. Nonnegative numbers of the form (r^n - n^r), for n,r > 1: A045575.

%Y 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 nonn

%O 1,1

%A _Matej Veselovac_, Sep 15 2020