A332436 The number of even numbers <= n of the smallest nonnegative reduced residue system modulo 2*n + 1, for n >= 0.

1, 0, 1, 1, 2, 2, 3, 2, 4, 4, 4, 5, 5, 4, 7, 7, 6, 6, 9, 6, 10, 10, 6, 11, 11, 8, 13, 10, 10, 14, 15, 8, 12, 16, 12, 17, 18, 10, 16, 19, 14, 20, 16, 14, 22, 18, 16, 18, 24, 14, 25, 25, 12, 26, 27, 18, 28, 22, 18, 24, 28, 20, 25, 31, 22, 32, 28, 18, 34, 34, 24

Offset: 0

Author

Wolfdieter Lang, Feb 29 2020

Keywords

• nonn
• easy

Comments

For the smallest positive reduced residue system modulo N see the array A038566. Here the nonnegative residue system [0, 1, ..., N-1] is considered, differing only for N = 1 from A038566, with [0] (instead of [1]).

This sequence gives the complement of A332435 (with 0 for n = 0 included) relative to the number of positive numbers <= n of the smallest nonnegative reduced residue system modulo (2*n+1). Thus a(n) + A332435(n) = phi(n)/2, for n >= 1, with phi = A000010. For n = 0 one has 1 + 0 = 1.

a(n) gives also the number of even numbers appearing in the complete modified doubling sequence system (name it MDS(b)), for b = 2*n + 1, with n >= 1, proposed in a comment from Gary W. Adamson, Aug 24 2019, in the example section of A135303 for prime b.

Examples

			n = 4, b = 9: the even numbers <= 4 in RRS(9) := [1, 2, 4, 5, 7, 8] are {2, 4}, hence a(4) = 2.
The complete MDS(9) system has one cycles of length 3: Cy*(9, 1) = (2, 4, 1), with the even numbers {2, 4}.
n = 8, b = 17: the even numbers <= 8 in RRS(17) := [1, 2, ..., 16] are {2, 4, 6 ,8}, hence a(8) = 4.
The complete MDS(17) system has two cycles of length 4: Cy*(17, 1) = (2, 4, 8, 1) and Cy*(17, 2) = (6, 5, 7, 3) and the even numbers are {2, 4, 6 ,8}.
		

Crossrefs

Cf. A000010, A038566, A135303, A332435.

Formula

a(n) = A000010(n)/2 - A332435(n), for n >= 1, and a(0) = 1.