OFFSET
1,1
COMMENTS
a(n) with the decimal representation a(n) = d_1 d_2 ... d_k is a term iff there is a unique ordered set of nonnegative integers m_1, m_2, ..., m_k >= 0 such that a(n) = d_1^m_1 + ... + d_k^m_k.
For any such term, the exponential Diophantine equation d_1 d_2 ... d_k = d_1^m_1 + ... + d_k^m_k with respect to m_1, m_2, ..., m_k has a unique solution in nonnegative integers.
Different powers of duplicate digits allow for multiple representations after a permutation, so that terms with equal digits must have equal powers on those positions (contrary to the definition of A050240, where such representations are considered equivalent).
None of the digits in a(n) can be 0 or 1, as it results in multiple representations, which makes it a subsequence of A387032.
LINKS
Dmytro Inosov, Table of n, a(n) for n = 1..1257
EXAMPLE
2 = 2^1, which is the only possible representation (the same holds for any single-digit number from 2 to 9);
24 = 2^3 + 4^2;
254 = 2^7 + 5^3 + 4^0;
9963 = 9^3 + 9^3 + 6^5 + 3^6;
98764 = 9^2 + 8^5 + 7^3 + 6^2 + 4^8.
Counterexamples:
264 is not a term because it has two distinct representations (see A050240):
264 = 2^1 + 6^1 + 4^4 = 2^5 + 6^3 + 4^2.
224 is not a term because 224 = 2^5 + 2^7 + 4^3 = 2^7 + 2^5 + 4^3 (these two representations are considered equivalent in A050240 because of duplicate digits).
153 = 1^3 + 5^3 + 3^3 is a narcissistic number (A005188). It is not a term since an infinite number of other representations exist: 153 = 1^m + 5^3 + 3^3 for any m >= 0.
CROSSREFS
KEYWORD
base,nonn
AUTHOR
Dmytro Inosov, Sep 15 2025
STATUS
approved