A388297 The smallest term in A061862 with exactly n distinct partitions into a sum of nonnegative powers of its digits.

2, 264, 739, 2334, 2332, 22864, 22953, 32786, 23845, 23852, 32842, 222933, 262249, 262287, 236493, 262254, 262282, 262223, 262498, 262234, 262283, 262248, 262458, 524328, 262243, 2223498, 262284, 284224, 262443, 2223854, 2223654, 2228434, 262442, 524322, 524382, 2228263

Offset

1,1

Comments

Any term in A061862 with the decimal representation d_1 d_2 ... d_k has at least one ordered set of nonnegative integers m_1, m_2, ..., m_k >= 0 such that d_1 d_2 ... d_k = d_1^m_1 + ... + d_k^m_k. a(n) gives the smallest term that has exactly n such partitions, not counting different powers of duplicate digits as distinct, or -1 if no such term exists.

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

Distinct digits raised to distinct powers are considered distinct even if they contribute equally to the sum. For example, 2^4 + 4^3 = 2^6 + 4^2 are counted as two distinct partitions.

None of the terms contains 0's or 1's among its digits because any such integer has either zero or infinite number of partitions. Therefore every term is also contained in A387032.

a(68) > 10^7.

Other known terms < 10^7: a(70) = 5243422; a(71) = 4228249; a(72) = 4792332; a(73) = 2228243; a(74) = 2228244; a(80) = 2228394; a(106) = 4228225; a(109) = 4228224.

Links

Dmytro Inosov, Table of n, a(n) for n = 1..67

Formula

a(1) = A388144(1).

a(2) = A050240(1).

Example

a(1) = 2 = 2^1 (unique partition).
a(2) = 264
     = 2^1 + 6^1 + 4^4
     = 2^5 + 6^3 + 4^2 (2 distinct partitions).
a(3) = 739
     = 7^0 + 3^2 + 9^3
     = 7^0 + 3^6 + 9^1
     = 7^1 + 3^1 + 9^3 (3 distinct partitions).
Note that here 3^6 + 9^1 = 3^2 + 9^3 are counted as distinct.
a(4) = 2334
     = 2^1 + 3^4 + 3^7 + 4^3
     = 2^7 + 3^1 + 3^7 + 4^2
     = 2^11 + 3^1 + 3^3 + 4^4
     = 2^11 + 3^3 + 3^5 + 4^2 (4 distinct partitions).
a(5) = 2332
     = 2^3 + 3^2 + 3^7 + 2^7
     = 2^4 + 3^0 + 3^7 + 2^7
     = 2^5 + 3^2 + 3^5 + 2^11
     = 2^5 + 3^4 + 3^7 + 2^5
     = 2^8 + 3^0 + 3^3 + 2^11 (5 distinct partitions).

Crossrefs

Cf. A003321, A005188, A007532, A023052, A050240, A061862, A134703, A384099, A388144.

Sequence in context: A103029 A122862 A236437 * A137105 A216984 A007082

Adjacent sequences: A388294 A388295 A388296 * A388298 A388299 A388300

Keyword

nonn,base

Author

Dmytro Inosov, Sep 16 2025

Status

approved