A349553 a(n) is the least k such that some halving partition of n has exactly k parts.

1, 5, 11, 19

Offset

1,2

Comments

Let S(m) = {f(m/2)), c(m/2) : m >= 1}, where f = floor and c = ceiling. A halving partition of n is a partition p(1) + p(2) + ... + p(k) of n such that p(1) is in S(n) and p(i) is in S(p(i-1)) for i = 1, 2, ..., k . The basic idea is that each term after the first is about half of the preceding term. (See A349552.)

Links

Table of n, a(n) for n=1..4.

Example

Let f = floor and c = ceiling.
a(1) counts this partition: c(1/2) = 1.
a(2) = 5 counts these partitions:
c(5/2) + c(3/2) = 5;
c(5/2) + f(3/2) + c(1/2) = 3 + 1 + 1 = 5.
a(3) = 11 counts these partitions:
c(11/2) + f(6/2) + c(3/2) = 6 + 3 + 2 = 11;
c(11/2) + f(6/2) + f(3/2) + c(1/2) = 6 + 3 + 1 + 1 = 11;
f(11/2) + c(5/2) + c(3/2) + f(2/2) = 5 + 3 + 2 + 1 = 11.
a(4) = 19 counts these partitions:
c(19/2) + f(10/2) + c(5/2) + f(3/2) = 10 + 5 + 3 + 1 = 19;
c(19/2) + f(10/2) + f(5/2) + f(2/2) + c(1/2) = 10 + 5 + 2 + 1 + 1 = 19;
f(19/2) + c(9/2) + c(5/2) + c(3/2) = 9 + 5 + 3 + 2 = 19;  f(19/2) + c(9/2) + c(5/2) + f(3/2) + c(1/2) = 9 + 5 + 3 + 1 + 1 = 19.

Crossrefs

Cf. A349552.

Sequence in context: A048253 A102174 A140515 * A056996 A102184 A290751

Adjacent sequences: A349550 A349551 A349552 * A349554 A349555 A349556

Keyword

nonn,more

Author

Clark Kimberling, Dec 26 2021

Status

approved