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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.

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

Original entry on oeis.org

1422, 1464, 1554, 2612, 3127, 4481, 5644, 16122, 68521, 77129, 82583, 1065585, 4227140, 6164560
Offset: 1

Views

Author

Matej Veselovac, Sep 28 2020

Keywords

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.

Examples

			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

Links

Crossrefs

Cf. A337670, A005188 (perfect digital invariants), perfect powers: A001597, A072103.
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