A339183 Number of partitions of n into two parts such that the smaller part is a nonzero square.

0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7

Offset: 0

Author

Wesley Ivan Hurt, Nov 26 2020

Keywords

• nonn
• easy

Comments

Every nonnegative integer k appears in blocks of 4k+2 k's.

Examples

			a(7) = 1; The partitions of 7 into two parts are (6,1), (5,2) and (4,3). Since 1 is the only nonzero square appearing among the smallest parts, a(7) = 1.
a(8) = 2; The partitions of 8 into two parts are (7,1), (6,2), (5,3) and  (4,4). Since 1 and 4 are the only nonzero squares appearing among the smallest parts, a(8) = 2.
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010052, A339184 (larger part is a nonzero square), A339186 (total number of nonzero squares).

Cf. A001105, A016825.

Programs

Mathematica

Table[Sum[Floor[Sqrt[i]] - Floor[Sqrt[i - 1]] , {i, Floor[n/2]}], {n, 0, 120}]

Python

from math import isqrt
def A339183(n): return isqrt(n>>1) # Chai Wah Wu, Oct 07 2024

Formula

a(n) = Sum_{i=1..floor(n/2)} c(i), where c is the square characteristic (A010052).

a(n) = A339186(n) - A339184(n).

a(n) = floor(sqrt(n/2)), for n >= 0. [See the comment, and {4*k+2 = A016825 (k)} is the first difference sequence of the sequence b(k) = 2*k^2 = A001105(k) for which sqrt(b(k)/2) becomes an integer.] - Wolfdieter Lang, Dec 07 2020