A383037 a(n) is the excess of composites over primes in the first n odd positive integers.

0, -1, -2, -3, -2, -3, -4, -3, -4, -5, -4, -5, -4, -3, -4, -5, -4, -3, -4, -3, -4, -5, -4, -5, -4, -3, -4, -3, -2, -3, -4, -3, -2, -3, -2, -3, -4, -3, -2, -3, -2, -3, -2, -1, -2, -1, 0, 1, 0, 1, 0, -1, 0, -1, -2, -1, -2, -1, 0, 1, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 4

Offset: 1

Author

Felix Huber, Apr 19 2025

Keywords

• sign
• easy

Examples

			Of the first 5 odd positive integers (1, 3, 5, 7, 9), one (9) is a composite and three (3, 5, 7) are primes, so a(5) = 1 - 3 = -2.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Odd Number

Crossrefs

Cf. A000720, A005408, A065091, A071904, A072731, A097454, A118777, A210469.

Programs

Maple

A383037:=n->n-NumberTheory:-pi(2*n)*2+1;seq(A383037(n),n=1..71);

Mathematica

a[n_]:=n - 2*PrimePi[2*n] + 1; Array[a,71] (* Stefano Spezia, Apr 20 2025 *)

Formula

a(n) = n - 2*pi(2*n) + 1.

a(n) = A210469(n) - pi(2*n) + 1 = A210469(n) - A000720(2*n) + 1 = for n > 1.

a(n) = A118777(2*n-1) - n + 1 for n > 1.

a(n) = A097454(2*n-1) - n + 2 for n > 1.

a(n) = A072731(2*n-1) - n + 3 for n > 1.