A383037 -id:A383037 - cp's OEIS frontend

A383259 a(n) is the excess of even composites over odd composites in the first n positive integers.

0, 0, 0, 1, 1, 2, 2, 3, 2, 3, 3, 4, 4, 5, 4, 5, 5, 6, 6, 7, 6, 7, 7, 8, 7, 8, 7, 8, 8, 9, 9, 10, 9, 10, 9, 10, 10, 11, 10, 11, 11, 12, 12, 13, 12, 13, 13, 14, 13, 14, 13, 14, 14, 15, 14, 15, 14, 15, 15, 16, 16, 17, 16, 17, 16, 17, 17, 18, 17, 18, 18, 19, 19, 20

Offset: 1

Felix Huber, Apr 24 2025

			Of the first 9 positive integers are three (4, 6, 8) even and one (9) is odd, so a(9) = 3 - 1 = 2.

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

Cf. A000034, A000720, A002808, A066247, A071904, A383037.

A383259:=n->`if`(n=1,0,NumberTheory:-pi(n)-(n mod 2)-1);seq(A383259(n),n=1..74);

a[1]:=0; a[n_]:=PrimePi[n]- Mod[n,2] - 1; Array[a,74] (* Stefano Spezia, Apr 25 2025 *)

a(n) = pi(n) - (n mod 2) - 1 = A000720(n) - A000034(n) for n > 1.

abs(a(n)-a(n-1)) = A066247(n).

A381251 a(n) is the number of ways to write prime(n) as a sum of distinct composites.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 4, 10, 14, 27, 40, 52, 74, 133, 229, 276, 457, 626, 744, 1189, 1599, 2498, 4450, 5862, 6752, 8835, 10139, 13189, 32481, 41614, 60099, 67900, 122825, 138101, 195147, 274193, 342783, 477381, 661502, 736865, 1252245, 1390615, 1711496, 1897886

Offset: 1

Felix Huber, Apr 19 2025

nonn

Subsequence of A204389.

All sums have an odd number of odd composite summands (A071904).

			a(8) = 3 because prime(8) = 19 can be written in 3 ways as a sum of distinct composites: 19 = 9 + 10 = 9 + 4 + 6 = 15 + 4.

Felix Huber, Table of n, a(n) for n = 1..4000

Cf. A000040, A002808, A053872, A071904, A204389, A383037.

P:= [seq(ithprime(i),i=1..100)]:
C:= {$2..P[-1]} minus convert(P,set):
G:= mul(1+x^c,c=C):
seq(coeff(G,x,P[i]),i=1..100); # Robert Israel, Apr 22 2025

a(n) = A204389(A000040(n)). - R. J. Mathar, Apr 22 2025

cp's OEIS Frontend

A383259 a(n) is the excess of even composites over odd composites in the first n positive integers.

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