A339183 -id:A339183 - cp's OEIS frontend

A183301 Complement of A014105.

1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

Clark Kimberling, Jan 03 2011

Cf. A014105, A339183.

a=2; b=1;
F[n_]:=a*n^2+b*n;
R[n_]:=(n/a+((b-1)/(2a))^2)^(1/2);
G[n_]:=n-1+Ceiling[R[n]-(b-1)/(2a)];
Table[G[n], {n,100}]

from math import isqrt
def A183301(n): return n+isqrt(n-1>>1) # Chai Wah Wu, Nov 04 2024

a(n) = n + floor(sqrt((n-1)/2)) = n + A339183(n-1). - Aaron J Grech, Jul 30 2024

A339184 Number of partitions of n into two parts such that the larger part is a nonzero square.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Wesley Ivan Hurt, Nov 26 2020

			a(8) = 1; The partitions of 8 into 2 parts are (7,1), (6,2), (5,3) and (4,4). Since 4 is the only nonzero square appearing as a largest part, a(8) = 1.
a(9) = 0; The partitions of 9 into 2 parts are (8,1), (7,2), (6,3) and (5,4). Since there are no nonzero squares among the largest parts, a(9) = 0.

Index entries for sequences related to partitions

Cf. A010052, A339183 (smaller part is a nonzero square), A339186 (total nonzero squares).

Table[Sum[Floor[Sqrt[n - i]] - Floor[Sqrt[n - i - 1]] , {i, Floor[n/2]}], {n, 0, 100}]

a(n) = Sum_{i=1..floor(n/2)} c(n-i), where c is the square characteristic (A010052).
a(n) = Sum_{i=floor((n-1)/2)..n-2} c(i+1), where c is the square characteristic (A010052).
a(n) = A339186(n) - A339183(n).

A339186 Total number of nonzero squares in the partitions of n into 2 parts.

Original entry on oeis.org

0, 0, 2, 1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Wesley Ivan Hurt, Nov 26 2020

			a(8) = 3; The partitions of 8 into two parts are (7,1), (6,2), (5,3) and (4,4). There are 3 total nonzero squares among the parts (namely 1, 4 and 4 ), so a(8) = 3.
a(9) = 2; The partitions of 9 into two parts are (8,1), (7,2), (6,3) and (5,4). Since 1 and 4 are the only nonzero squares among all parts, a(9) = 2.

Index entries for sequences related to partitions

Cf. A339183 (number of smaller parts), A339184 (number of larger parts).

Table[Sum[Floor[Sqrt[i]] - Floor[Sqrt[i - 1]] + Floor[Sqrt[n - i]] - Floor[Sqrt[n - i - 1]], {i, Floor[n/2]}], {n, 0, 100}]

a(n) = A339184(n) + A339183(n).

cp's OEIS Frontend

A183301 Complement of A014105.

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A339184 Number of partitions of n into two parts such that the larger part is a nonzero square.

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A339186 Total number of nonzero squares in the partitions of n into 2 parts.

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