A380464 Integers k such that A005245(m*k) < A005245(k) for some m.

1499, 1823, 3767, 5468, 5469, 13163, 13487, 16403, 16407, 20507, 25799, 28607, 30713, 30983, 32828, 36383

Offset

1,1

Comments

A005245(n) is the integer complexity of n, which is the least number of copies of 1 needed to express n with addition and multiplication (and legal nestings of brackets). Although there are logarithmic upper and lower bounds for A005245(n), there are known instances such that it is not the case that A005245(n) <= A005245(m*n) for each of m = 2 and m = 3 (see the Examples below).

Is this integer sequence infinite? This is an open problem.

Links

Table of n, a(n) for n=1..16.

Harry Altman, Integer Complexity: Algorithms and Computational Results, Integers, 18 (2018), A45.

Harry Altman, Integer Complexity: The Integer Defect, arXiv:1804.07446 [math.NT], 2018; Mosc. J. Comb. Number Theory, 8 (2019), 193-217.

Formula

Integers k such that A005245(k) > min{A005245(k), A005245(2*k), ..., A005245((k-1)*k)}.

Example

We find that A005245(1499) = 25 and that A005245(2*1499) = 24, and 1499 is the smallest number k such that A005245(m*k) < A005245(k), so that a(1) = 1499.
In the given Altman references, it is noted that the integer k = 4721323 is such that A005245(3*k) < A005245(k), so 4721323 is included in this sequence.

Crossrefs

Cf. A005245, A195101, A350723, A351467.

Sequence in context: A345781 A328974 A236732 * A184078 A098191 A253509

Adjacent sequences: A380461 A380462 A380463 * A380465 A380466 A380467

Keyword

nonn,more

Author

John M. Campbell, Jun 22 2025

Status

approved