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A337671 Subsequence of A337670 in which there are at most five terms in the sum. 2
1422, 1464, 1554, 2612, 3127, 4481, 5644, 16122, 68521, 77129, 82583, 1065585, 4227140, 6164560 (list; graph; refs; listen; history; text; internal format)
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? Are there infinitely many terms whose sum consists of exactly 5 powers?
If it exists, a(15) > 10^20.
LINKS
Math StackExchange, Base-Exponent Invariants, 2020.
Eric Weisstein's World of Mathematics, Perfect Power.
EXAMPLE
a(1) = 1422 = 2^5 + 2^7 + 5^3 + 5^4 + 2^9 = 5^2 + 7^2 + 3^5 + 4^5 + 9^2
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 + 8^2 + 5^4 + 3^6 = 3^2 + 7^2 + 2^8 + 4^5 + 6^3
a(4) = 2612 = 2^5 + 2^6 + 5^3 + 7^3 + 2^11 = 5^2 + 6^2 + 3^5 + 3^7 + 11^2
a(5) = 3127 = 2^3 + 2^9 + 6^3 + 7^3 + 2^11 = 3^2 + 9^2 + 3^6 + 3^7 + 11^2
a(6) = 4481 = 2^6 + 7^2 + 2^10 + 2^11 + 6^4 = 6^2 + 2^7 + 10^2 + 11^2 + 4^6
a(7) = 5644 = 4^5 + 9^2 + 10^2 + 7^3 + 4^6 = 5^4 + 2^9 + 2^10 + 3^7 + 6^4
a(8) = 16122 = 2^3 + 4^3 + 2^8 + 5^6 + 13^2 = 3^2 + 3^4 + 8^2 + 6^5 + 2^13
a(9) = 68521 = 2^8 + 4^5 + 4^6 + 3^10 + 8^4 = 8^2 + 5^4 + 6^4 + 10^3 + 4^8
a(10) = 77129 = 4^6 + 2^12 + 7^4 + 10^3 + 2^16 = 6^4 + 12^2 + 4^7 + 3^10 + 16^2
a(11) = 82583 = 2^5 + 4^3 + 12^2 + 7^5 + 2^16 = 5^2 + 3^4 + 2^12 + 5^7 + 16^2
a(12) = 1065585 = 2^9 + 2^12 + 7^4 + 10^4 + 2^20 = 9^2 + 12^2 + 4^7 + 4^10 + 20^2
a(13) = 4227140 = 5^6 + 13^2 + 7^4 + 11^4 + 2^22 = 6^5 + 2^13 + 4^7 + 4^11 + 22^2
a(14) = 6164560 = 5^7 + 2^18 + 9^5 + 21^2 + 7^8 = 7^5 + 18^2 + 5^9 + 2^21 + 8^7
CROSSREFS
Cf. A337670, A005188 (perfect digital invariants), perfect powers: A001597, A072103.
Sequence in context: A252114 A364125 A293093 * A237960 A023937 A225567
KEYWORD
nonn,hard,more
AUTHOR
Matej Veselovac, Sep 28 2020
STATUS
approved

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Last modified April 14 20:39 EDT 2024. Contains 371667 sequences. (Running on oeis4.)