A210469 -id:A210469 - cp's OEIS frontend

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

Felix Huber, Apr 19 2025

			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.

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

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

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

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

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.

A224710 The number of unordered partitions {a,b} of 2n-1 such that a and b are composite.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 7, 8, 8, 8, 9, 9, 10, 11, 11, 12, 13, 13, 13, 14, 15, 15, 16, 16, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 28, 29, 30, 31, 32, 33, 33, 34, 34, 35, 36, 36

Offset: 1

J. Stauduhar, Apr 16 2013

nonn

Except for the initial terms, the same sequence as A210469.

			n=7: 13 has a unique representation as the sum of two composite numbers, namely 13 = 4+9, so a(7)=1.

J. Stauduhar, Table of n, a(n) for n = 1..10000

Subsequence of A224708. Cf. A210469.

Table[Length@ Select[IntegerPartitions[2 n - 1, {2}] /. n_Integer /; ! CompositeQ@ n -> Nothing, Length@ # == 2 &], {n, 71}] (* Version 10.2, or *)
Table[If[n == 1, 0, n - 2 - PrimePi[2 n - 4]], {n, 71}] (* Michael De Vlieger, May 03 2016 *)

a(n) = n - 2 - primepi(2n-4) for n>1. - Anthony Browne, May 03 2016

a(A104275(n+2) + 1) = n. - Anthony Browne, May 25 2016

A308754 a(0) = 0, a(n) = a(n-1) + 1 if 2*n + 3 is prime, otherwise a(n) = a(n-1).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 9, 9, 9, 10, 10, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 16, 17, 17, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 28, 28, 28, 28, 28, 28, 29, 29, 30, 30, 30, 31

Offset: 0

Carlos Fernandez Rivero, Jun 22 2019

It appears that A000040(a(n)) ~ 2*n as n tends to infinity. (See Mar 12 2012 note from Vladimir Shevelev in A060308.)

			a(0) = 0 (by definition).
a(1) = 1 = a(0) + 1, because 2*1 + 3 is prime;
a(2) = 2 = a(1) + 1, because 2*2 + 3 is prime;
a(3) = 2 = a(2),     because 2*3 + 3 is not prime;
a(4) = 3 = a(3) + 1, because 2*4 + 3 is prime.

Carlos Fernandez Rivero, Table of n, a(n) for n = 0..10000
Carlos Fernandez Rivero, Plots of a(n) for n = 0..200 and n = 0..10000

Cf. A000040, A000720, A060265, A060308, A085090, A099801, A101264.

' p(n) contains the prime sequence except for 2. p(0)=3
' output in the a(n) sequence for 0 <= n <= maxterm
ip = -1
For n = 0 To maxterm
   If (2 * n + 3) = p(ip+1) Then
      ip = ip + 1
   End If
   a(n) = ip
Next n

[#PrimesUpTo(2*n + 4) - 2: n in [0..80] ]; // Vincenzo Librandi, Aug 01 2019

a[0] = 0; a[n_] := a[n] = a[n - 1] + Boole@PrimeQ[2 n + 3]; Array[a, 100, 0] (* Amiram Eldar, Jul 06 2019 *)

a(n) = a(n-1) + A101264(n+1), n > 0.
a(n) = A000720(2 * (n+2)) - 2.
a(n) = A099801(n+1) - 2.
a(n) = n - A210469(n+2).
A000040(a(n) + 2) = A060265(n+2).
A000040(a(n) + 2) = A060308(n+2).
A000040(a(n) + 2) = A085090(n+2), if 2*n + 3 is prime, otherwise 0.

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

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A224710 The number of unordered partitions {a,b} of 2n-1 such that a and b are composite.

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A308754 a(0) = 0, a(n) = a(n-1) + 1 if 2*n + 3 is prime, otherwise a(n) = a(n-1).

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