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

Author

John M. Campbell, Jun 22 2025

Keywords

• nonn
• more

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.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A005245, A195101, A350723, A351467.

Formula

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