A384099 Number of ways A061862(n) can be represented as a sum of nonnegative powers of its digits, or -1 if this number is infinite.

-1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 6, 2, 1, 1, -1, -1, 2, 4, 1, 2, 1, 2, 1, 1, -1, 2, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 1, 2, 1, 1, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset

1,20

Comments

By definition, for any A061862(n) with the decimal representation d_1 d_2 ... d_k there exists at least one ordered set of nonnegative integers m_1, m_2, ..., m_k >= 0 such that A061862(n) = d_1^m_1 + ... + d_k^m_k. a(n) gives the number of distinct ordered sets {m_1, m_2, ..., m_k} that satisfy this equation. Whenever A061862(n) contains a 0 or 1 among its digits, the number of such sets is infinite, in that case a(n) is defined to be -1.

a(n) gives the number of solutions to 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 in nonnegative integers.

Zero digits do not contribute to the sum (using 0^0 = 1 is forbidden, as per A061862).

a(n) != 0, because all terms in A061862 have at least one such solution by definition.

a(n) > 0 iff A061862(n) is a term in A387032.

a(n) = 1 iff A061862(n) is a term in A388144.

If A061862(n) is a term in A050240, a(n) >= 2, the equality holds if both representations have equal powers on positions corresponding to repeating digits.

Links

Dmytro Inosov, Table of n, a(n) for n = 1..10000

Dmytro Inosov, Table of n, A061862(n), A384099(n), and the number of nonequivalent representations not counting different powers of duplicate digits as distinct for n = 1..10000

Crossrefs

Cf. A003321, A005188, A007532, A023052, A050240, A061862, A134703, A388144.

Sequence in context: A030410 A085301 A337565 * A138385 A030614 A381733

Adjacent sequences: A384096 A384097 A384098 * A384100 A384101 A384102

Keyword

sign,base

Author

Dmytro Inosov, Sep 16 2025

Status

approved