OFFSET
1,1
COMMENTS
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.
If it exists, what is the smallest term whose sum consists of exactly 2, 3 or 4 powers?
From Matej Veselovac, Jun 20 2026: (Start)
Call a(n) an "order-k genus-g" term if its k pairs {b_i,e_i} (x_i=min(b_i,e_i), y_i=max(b_i,e_i), 1<x_i<y_i) can be split into two groups of sizes k_1 and k_2 (k_1+k_2=k, g=min(k_1,k_2)) such that Sum_{i in group 1} (x_i^y_i - y_i^x_i) = Sum_{j in group 2} (x_j^y_j - y_j^x_j).
In other words, write each pair as x^y - y^x, 1<x<y. Recognize genus-1 terms by all pairs but one falling on the same side of the equation.
a(2) -> 3^6 - 6^3 = (2^5 - 5^2 ) + (4^5 - 5^4 ) + (2^6 - 6^2 ) + (2^7 - 7^2 )
a(9) -> 4^8 - 8^4 = (4^5 - 5^4 ) + (4^6 - 6^4 ) + (2^8 - 8^2 ) + (3^10 - 10^3 )
a(15) -> 2^513 - 513^2 = (2^3 - 3^2 ) + (2^32 - 32^2 ) + (4^256 - 256^4 ) + (2^512 - 512^2 )
a(17)? -> 2^9216 - 9216^2 = (4^96 - 96^4) + (32^64 - 64^32) + (32^1024 - 1024^32) + (512^1024 - 1024^512)
Is the next term the 2775-digit example? Exhaustive search up to 10^700 found no additional terms.
There is an infinite family of order-5 genus-2 solutions N(p) given by:
N(p) = (p)^(8p^(3p)) + (p^(3p))^(p^8) + (p^9)^(p^3) + (9p^2)^(p^p) + (2p^p)^(3p)
= (8p^(3p))^(p) + (p^8)^(p^(3p)) + (p^3)^(p^9) + (p^p)^(9p^2) + (3p)^(2p^p)
for every p>1. The only exception is p=3 when the solution is order-5 genus-1. (End)
LINKS
Matej Veselovac, Table of n, a(n) for n = 1..16
Math StackExchange, Base-Exponent Invariants, 2020.
Math StackExchange, Sums of four a^b-b^a numbers, 2026.
Matej Veselovac, Python program for A337671, 2020.
André Weis, Python program for a(15), 2026.
Eric Weisstein's World of Mathematics, Perfect Power.
EXAMPLE
a(1) = 1422 = 2^5 + 2^7 + 2^9 + 5^3 + 5^4 = 5^2 + 7^2 + 9^2 + 3^5 + 4^5
a(2) = 1464 = 2^5 + 2^6 + 2^7 + 4^5 + 6^3 = 5^2 + 6^2 + 7^2 + 5^4 + 3^6
a(3) = 1554 = 2^3 + 2^7 + 3^6 + 5^4 + 8^2 = 3^2 + 7^2 + 6^3 + 4^5 + 2^8
a(4) = 2612 = 2^5 + 2^6 + 2^11 + 5^3 + 7^3 = 5^2 + 6^2 + 11^2 + 3^5 + 3^7
a(5) = 3127 = 2^3 + 2^9 + 2^11 + 6^3 + 7^3 = 3^2 + 9^2 + 11^2 + 3^6 + 3^7
a(6) = 4481 = 2^6 + 2^10 + 2^11 + 6^4 + 7^2 = 6^2 + 10^2 + 11^2 + 4^6 + 2^7
a(7) = 5644 = 4^5 + 4^6 + 7^3 + 9^2 + 10^2 = 5^4 + 6^4 + 3^7 + 2^9 + 2^10
a(8) = 16122 = 2^3 + 2^8 + 4^3 + 5^6 + 13^2 = 3^2 + 8^2 + 3^4 + 6^5 + 2^13
a(9) = 68521 = 2^8 + 3^10 + 4^5 + 4^6 + 8^4 = 8^2 + 10^3 + 5^4 + 6^4 + 4^8
a(10) = 77129 = 2^12 + 2^16 + 4^6 + 7^4 + 10^3 = 12^2 + 16^2 + 6^4 + 4^7 + 3^10
a(11) = 82583 = 2^5 + 2^16 + 4^3 + 7^5 + 12^2 = 5^2 + 16^2 + 3^4 + 5^7 + 2^12
a(12) = 1065585 = 2^9 + 2^12 + 2^20 + 7^4 + 10^4 = 9^2 + 12^2 + 20^2 + 4^7 + 4^10
a(13) = 4227140 = 2^22 + 5^6 + 7^4 + 11^4 + 13^2 = 22^2 + 6^5 + 4^7 + 4^11 + 2^13
a(14) = 6164560 = 2^18 + 5^7 + 7^8 + 9^5 + 21^2 = 18^2 + 7^5 + 8^7 + 5^9 + 2^21
a(15) = (155) = 2^3 + 2^32 + 2^512 + 4^256 + 513^2 = 3^2 + 32^2 + 512^2 + 256^4 + 2^513
a(16) = (463) = 2^512 + 8^6 + 36^4 + 64^256 + 512^8 = 512^2 + 6^8 + 4^36 + 256^64 + 8^512
a(17)?= (2775) = 2^9216 + 96^4 + 64^32 + 1024^32 + 1024^512 = 9216^2 + 4^96 + 32^64 + 32^1024 + 512^1024
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Matej Veselovac, Sep 28 2020
STATUS
approved