cp's OEIS Frontend

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

%I A339183 #23 Oct 07 2024 15:28:35
%S A339183 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,
%T A339183 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,
%U A339183 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
%N A339183 Number of partitions of n into two parts such that the smaller part is a nonzero square.
%C A339183 Every nonnegative integer k appears in blocks of 4k+2 k's.
%H A339183 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A339183 a(n) = Sum_{i=1..floor(n/2)} c(i), where c is the square characteristic (A010052).
%F A339183 a(n) = A339186(n) - A339184(n).
%F A339183 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
%e A339183 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.
%e A339183 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.
%t A339183 Table[Sum[Floor[Sqrt[i]] - Floor[Sqrt[i - 1]] , {i, Floor[n/2]}], {n, 0, 120}]
%o A339183 (Python)
%o A339183 from math import isqrt
%o A339183 def A339183(n): return isqrt(n>>1) # _Chai Wah Wu_, Oct 07 2024
%Y A339183 Cf. A010052, A339184 (larger part is a nonzero square), A339186 (total number of nonzero squares).
%Y A339183 Cf. A001105, A016825.
%K A339183 nonn,easy
%O A339183 0,9
%A A339183 _Wesley Ivan Hurt_, Nov 26 2020