This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A337671 #19 Nov 29 2020 14:14:53
%S A337671 1422,1464,1554,2612,3127,4481,5644,16122,68521,77129,82583,1065585,
%T A337671 4227140,6164560
%N A337671 Subsequence of A337670 in which there are at most five terms in the sum.
%C A337671 Number m is in the sequence if there exists a set of unordered {base, exponent} pairs {{b_1, e_1}, ..., {b_k, e_k}}, k <= 5, representing non-commutative perfect powers b_i^e_i != e_i^b_i, b_i > 1, e_i > 1, whose sum equals m = Sum_{i=1..k} b_i^e_i = Sum_{i=1..k} e_i^b_i.
%C A337671 If it exists, what is the smallest term whose sum consists of exactly 2, 3 or 4 powers? Are there infinitely many terms whose sum consists of exactly 5 powers?
%C A337671 If it exists, a(15) > 10^20.
%H A337671 Math StackExchange, <a href="https://math.stackexchange.com/q/3795656/318073">Base-Exponent Invariants</a>, 2020.
%H A337671 Matej Veselovac, <a href="/A337671/a337671.txt">PYTHON program for A337671</a>
%H A337671 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>.
%e A337671 a(1) = 1422 = 2^5 + 2^7 + 5^3 + 5^4 + 2^9 = 5^2 + 7^2 + 3^5 + 4^5 + 9^2
%e A337671 a(2) = 1464 = 2^5 + 2^6 + 2^7 + 4^5 + 6^3 = 5^2 + 6^2 + 7^2 + 5^4 + 3^6
%e A337671 a(3) = 1554 = 2^3 + 2^7 + 8^2 + 5^4 + 3^6 = 3^2 + 7^2 + 2^8 + 4^5 + 6^3
%e A337671 a(4) = 2612 = 2^5 + 2^6 + 5^3 + 7^3 + 2^11 = 5^2 + 6^2 + 3^5 + 3^7 + 11^2
%e A337671 a(5) = 3127 = 2^3 + 2^9 + 6^3 + 7^3 + 2^11 = 3^2 + 9^2 + 3^6 + 3^7 + 11^2
%e A337671 a(6) = 4481 = 2^6 + 7^2 + 2^10 + 2^11 + 6^4 = 6^2 + 2^7 + 10^2 + 11^2 + 4^6
%e A337671 a(7) = 5644 = 4^5 + 9^2 + 10^2 + 7^3 + 4^6 = 5^4 + 2^9 + 2^10 + 3^7 + 6^4
%e A337671 a(8) = 16122 = 2^3 + 4^3 + 2^8 + 5^6 + 13^2 = 3^2 + 3^4 + 8^2 + 6^5 + 2^13
%e A337671 a(9) = 68521 = 2^8 + 4^5 + 4^6 + 3^10 + 8^4 = 8^2 + 5^4 + 6^4 + 10^3 + 4^8
%e A337671 a(10) = 77129 = 4^6 + 2^12 + 7^4 + 10^3 + 2^16 = 6^4 + 12^2 + 4^7 + 3^10 + 16^2
%e A337671 a(11) = 82583 = 2^5 + 4^3 + 12^2 + 7^5 + 2^16 = 5^2 + 3^4 + 2^12 + 5^7 + 16^2
%e A337671 a(12) = 1065585 = 2^9 + 2^12 + 7^4 + 10^4 + 2^20 = 9^2 + 12^2 + 4^7 + 4^10 + 20^2
%e A337671 a(13) = 4227140 = 5^6 + 13^2 + 7^4 + 11^4 + 2^22 = 6^5 + 2^13 + 4^7 + 4^11 + 22^2
%e A337671 a(14) = 6164560 = 5^7 + 2^18 + 9^5 + 21^2 + 7^8 = 7^5 + 18^2 + 5^9 + 2^21 + 8^7
%Y A337671 Cf. A337670, A005188 (perfect digital invariants), perfect powers: A001597, A072103.
%K A337671 nonn,hard,more
%O A337671 1,1
%A A337671 _Matej Veselovac_, Sep 28 2020