A337671 Subsequence of A337670 in which there are at most five terms in the sum.

1422, 1464, 1554, 2612, 3127, 4481, 5644, 16122, 68521, 77129, 82583, 1065585, 4227140, 6164560

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

Table of n, a(n) for n=1..14.

Math StackExchange, Base-Exponent Invariants, 2020.

Matej Veselovac, PYTHON program for A337671

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

Adjacent sequences: A337668 A337669 A337670 * A337672 A337673 A337674

Keyword

nonn,hard,more

Author

Matej Veselovac, Sep 28 2020

Status

approved