A224708 -id:A224708 - cp's OEIS frontend

A045917 From Goldbach problem: number of decompositions of 2n into unordered sums of two primes.

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3, 7, 9, 6, 5, 8, 7, 8, 11, 6, 5, 12, 4, 8, 11, 5, 8, 10, 5, 6, 13, 9, 6, 11, 7, 7, 14, 6, 8, 13, 5, 8, 11, 7, 9

Offset: 1

Felice Russo

Note that A002375 (which differs only at the n = 2 term) is the main entry for this sequence.
The graph of this sequence is called Goldbach's comet. - David W. Wilson, Mar 19 2012
This is the row length sequence of A182138, A184995 and A198292. - Jason Kimberley, Oct 03 2012
The Goldbach conjecture states that a(n) > 0 for n >= 2. - Wolfdieter Lang, May 14 2016
With the second Maple program, the command G(2n) yields all the unordered pairs of prime numbers having sum 2n; caveat: a pair {a,a} is listed as {a}. Example: G(26) yields {{13}, {3,23}, {7,19}}. The command G(100000) yields 810 pairs very fast. - Emeric Deutsch, Jan 03 2017
Conjecture: Let p denote any prime in any decomposition of 2n. 4 and 6 are the only numbers n such that 2n + p is prime for every p. - Ivan N. Ianakiev, Apr 06 2017
Conjecture: For all m >= 0, there exists at least one possible value of n such that a(n) = m. - Ahmad J. Masad, Jan 06 2018
The previous conjecture is related to the sequence A053033. - Ahmad J. Masad, Dec 09 2019
Conjecture: For each k >= 0, there exists a minimum sufficiently large number r that depends on k such that for each n >= r, a(n) > k. - Ahmad J. Masad, Jan 08 2020
Conjecture: If the previous conjecture is true, then for each m >= 0, the number of terms that are equal to (m+1) is larger than the number of terms that are equal to m. - Ahmad J. Masad, Jan 08 2020
Also, the number of equidistant prime pairs in Goldbach's Prime Triangle for integers n > 2. An equidistant prime pair is a pair of not necessarily different prime numbers (p1, p2) that have the same distance d >= 0 from an integer n, i.e., so that p1 = n - d and p2 = n + d. - Jörg Winkelmann, Mar 05 2025

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, MA, 1996, Chapter 12, pages 236-257.
H. Halberstam and H. E. Richert, 1974, "Sieve methods", Academic press, London, New York, San Francisco.

H. J. Smith, Table of n, a(n) for n = 1..20000
M. Herkommer, Goldbach Conjecture Research.
Eric Weisstein's World of Mathematics, Goldbach Partition.
Wikipedia, Goldbach's conjecture.
G. Xiao, WIMS server, Goldbach
Index entries for sequences related to Goldbach conjecture
J. Winkelmann, Goldbach's Prime Triangle — A Recreational Math Journey with an Introduction to Equidistant Primes.

Cf. A002375 (the main entry for this sequence (which differs only at the n=2 term)).
Cf. A023036 (first appearance of n), A000954 (last (assumed) appearance of n).
Cf. also A000040, A182138, A184995, A185297, A187129, A010051, A198292, A035026, A224708, A224709.
Cf. A064911, A105020.

a045917 n = sum $ map (a010051 . (2 * n -)) $ takeWhile (<= n) a000040_list
-- Reinhard Zumkeller, Sep 02 2013

[#RestrictedPartitions(2*n,2,Set(PrimesInInterval(1,2*n))):n in [1..100]]; // Marius A. Burtea, Jan 23 2020

A045917 := proc(n)
    local a,i ;
    a := 0 ;
    for i from 1 to n do
        if isprime(i) and isprime(2*n-i) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 01 2013
# second Maple program:
G := proc (n) local g, j: g := {}: for j from 2 to (1/2)*n do if isprime(j) and isprime(n-j) then g := `union`(g, {{n-j, j}}) end if end do: g end proc: seq(nops(G(2*n)), n = 1 .. 98); # Emeric Deutsch, Jan 03 2017

f[n_] := Length[Select[2n - Prime[Range[PrimePi[n]]], PrimeQ]]; Table[ f[n], {n, 100}] (* Paul Abbott, Jan 11 2005 *)
nn = 10^2; ps = Boole[PrimeQ[Range[1,2*nn,2]]]; Join[{0,1}, Table[Sum[ps[[i]] ps[[n-i+1]], {i, Ceiling[n/2]}], {n, 3, nn}]] (* T. D. Noe, Apr 13 2011 *)

a(n)=my(s);forprime(p=2,n,s+=isprime(2*n-p));s \\ Charles R Greathouse IV, Mar 27 2012

from sympy import isprime
def A045917(n):
    x = 0
    for i in range(2,n+1):
        if isprime(i) and isprime(2*n-i):
            x += 1
    return x # Chai Wah Wu, Feb 24 2015

From Halberstam and Richert: a(n) < (8+0(1))*c(n)*n/log(n)^2 where c(n) = Product_{p>2} (1 - 1/(p-1)^2)*Product_{p|n, p>2} (p-1)/(p-2). It is conjectured that the factor 8 can be replaced by 2. - Benoit Cloitre, May 16 2002
a(n) = ceiling(A035026(n) / 2) = (A035026(n) + A010051(n)) / 2.
a(n) = Sum_{i=2..n} floor(2/Omega(i*(2*n-i))). - Wesley Ivan Hurt, Jan 24 2013
a(n) = A224709(n) + (primepi(2n-2) - primepi(n-1)) + primepi(n) + 1 - n. - Anthony Browne, May 03 2016
a(n) = A224708(2n) - A224708(2n+1) + A010051(n). - Anthony Browne, Jun 26 2016
a(n) = Sum_{k=n*(n-1)/2+2..n*(n+1)/2} A064911(A105020(k-1)). - Wesley Ivan Hurt, Sep 11 2021
a(n) = omega(A362641(n)) = omega(A362640(n)). - Wesley Ivan Hurt, Apr 28 2023

A224712 The number of unordered partitions {a, b} of n such that a or b is composite and the other is prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 4, 2, 3, 2, 4, 2, 6, 2, 5, 3, 6, 3, 8, 2, 7, 4, 9, 5, 9, 3, 8, 6, 9, 4, 11, 3, 11, 8, 10, 6, 12, 4, 11, 7, 12, 7, 14, 4, 13, 7, 15, 9, 15, 5, 14, 10, 16, 9, 16, 4, 15, 12, 16, 8, 18, 6, 18, 14, 17, 9, 19, 7, 18, 11, 19, 11, 21

Offset: 1

J. Stauduhar, Apr 20 2013

			For n = 6, in the set {{5, 1}, {4, 2}, {3, 3}}, {4, 2} is the only partition that satisfies the requirements, so a(6) = 1.
For n = 9, we have partitions {6, 3} and {5, 4}, so a(9) = 2.

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

Cf. A062602 (allows 1 as well as composites), A224708 (a and b are both composite).

Table[Length[Select[Range[2, Floor[n/2]], (PrimeQ[#] && Not[PrimeQ[n - #]]) || (Not[PrimeQ[#]] && PrimeQ[n - #]) &]], {n, 80}] (* Alonso del Arte, Apr 21 2013 *)
Table[Count[IntegerPartitions[n,{2}],?(FreeQ[#,1]&&Total[Boole[ PrimeQ[ #]]] == 1&)],{n,80}] (* _Harvey P. Dale, Jul 21 2021 *)

a(n)=my(s);forprime(p=2,n-4,s+=!isprime(n-p));s \\ Charles R Greathouse IV, Apr 30 2013

A224709 The number of unordered partitions {a,b} of the even numbers 2n such that a and b are composite.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 4, 4, 4, 6, 5, 6, 8, 7, 8, 10, 8, 10, 12, 11, 11, 14, 13, 13, 16, 14, 15, 19, 15, 18, 20, 17, 20, 22, 20, 21, 24, 22, 22, 27, 23, 24, 30, 25, 26, 30, 27, 30, 33, 30, 30, 34, 32, 33, 37, 33, 33, 41, 33, 36, 42, 36, 40, 43, 39, 40, 44

Offset: 1

J. Stauduhar, Apr 16 2013

nonn

Conjecture: a(3n+9) > a(3n+8) and a(3n+10) < a(3n+9) for n>=1. - Anthony Browne, Jun 26 2016

			For n=6, 2*6=12 and the partitions of 12 are (1,11),(2,10),(3,9),(4,8),(5,7),(6,6). Of these, 2 are composite pairs, namely (4,8),(6,6) so a(6)=2.

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

Subsequence of A224708.

Cf. A010051.

Table[Count[Transpose@ {#, 2 n - #} &@ Range@ n, w_ /; Times @@ Boole@ Map[CompositeQ, w] > 0], {n, 69}] (* Michael De Vlieger, Jun 26 2016 *)

a(n) = sum(k=1, n-1, (1-isprime(k+1))*(1-isprime(2*n-k-1))); \\ Michel Marcus, Apr 11 2022

a(n) = Sum_{k=1..n-1} (1-A010051(k+1))(1-A010051(2n-k-1)). - Anthony Browne, Jun 26 2016

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

A024683 a(n) is the number of ways prime(n) is a sum of two composite numbers r,s satisfying r < s.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 6, 7, 8, 8, 10, 12, 13, 14, 15, 16, 17, 18, 20, 23, 24, 25, 25, 26, 26, 32, 33, 35, 36, 39, 40, 41, 43, 44, 46, 48, 49, 52, 53, 53, 54, 58, 63, 64, 65, 65, 67, 68, 71, 73, 75, 77, 78, 79, 80, 81, 84, 90, 91, 92, 92, 98, 100, 104, 105, 105, 107, 110, 112, 114

Offset: 1

Clark Kimberling

J. Stauduhar, Table of n, a(n) for n = 1..5000

Subsequence of A224708.

Cf. A000040, A066247.

z = 400; c = Select[Range[2, z], ! PrimeQ@# &];  (* A002808 *)
d = Select[Range[2, z], ! PrimeQ@# && OddQ@# &];  (* A014076 *)
a[n_] := Length[Intersection[c, Prime[n] - Select[d, # < Prime[n] &]]];
Table[a[n], {n, 1, 120}] (* A024683 *)
(* Clark Kimberling, Jul 21 2020 *)
Table[Count[IntegerPartitions[Prime[n],{2}],?(AllTrue[#,CompositeQ]&&#[[1]]!= #[[2]]&)],{n,80}] (* _Harvey P. Dale, Feb 24 2023 *)

a(n) = Sum_{i=4..floor((prime(n)-1)/2)} c(i) * c(prime(n)-i), where c is the characteristic function of composite numbers (A066247) and prime(n) is the n-th prime (A000040). - Wesley Ivan Hurt, Sep 08 2020

A171691 Number of unordered partitions {k1, k2} of n such that k1 and k2 are nonnegative nonprimes A141468.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 3, 1, 3, 2, 3, 3, 5, 2, 5, 3, 5, 4, 6, 3, 7, 5, 7, 5, 8, 5, 9, 6, 8, 7, 10, 7, 12, 7, 9, 9, 12, 8, 13, 9, 12, 10, 13, 9, 15, 11, 15, 11, 15, 11, 17, 13, 16, 13, 17, 13, 20, 14, 16, 15, 20, 15, 22, 15, 18, 17, 22, 16, 23, 17, 21, 18, 23, 18, 26, 18, 23

Offset: 1

Juri-Stepan Gerasimov, Dec 15 2009

nonn

			a(1) = 1 because 1 = 0 + 1.
a(2) = 1 because 2 = 1 + 1.
a(3) = 0.
a(4) = 1 because 4 = 0 + 4.
a(5) = 1 because 5 = 1 + 4.
a(6) = 1 because 6 = 0 + 6.
a(7) = 1 because 7 = 1 + 6.
a(8) = 2 because 8 = 0 + 8 = 4 + 4.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A062610, A141468, A224708.

a(n)={sum(i=0, n\2, (i<2 || !isprime(i)) && !isprime(n-i))} \\ Andrew Howroyd, Jan 05 2020

Name clarified and terms a(55) and beyond from Andrew Howroyd, Jan 05 2020

A339927 Number of partitions of n into two composite parts with the same number of divisors.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 2, 0, 2, 0, 2, 1, 1, 1, 2, 1, 0, 1, 2, 3, 3, 1, 4, 1, 2, 2, 4, 2, 1, 1, 4, 4, 3, 3, 4, 2, 2, 3, 7, 4, 3, 0, 4, 4, 5, 2, 5, 3, 1, 3, 7, 6, 3, 3, 6, 6, 5, 3, 6, 2, 6, 3, 11, 7, 2, 3, 4, 6, 5, 5, 8, 3, 4, 5, 10, 4, 4, 3, 7, 5, 7, 7, 7

Offset: 1

Wesley Ivan Hurt, Dec 23 2020

nonn

			a(18) = 2; 18 has two partitions into two composite parts that have the same number of divisors, (10,8) and (9,9).

Index entries for sequences related to partitions

Cf. A000005, A066247, A224708.

Table[Sum[KroneckerDelta[DivisorSigma[0, i], DivisorSigma[0, n - i]] (1 - PrimePi[i] + PrimePi[i - 1]) (1 - PrimePi[n - i] + PrimePi[n - i - 1]), {i, 2, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[#,CompositeQ]&&Length[Union[ DivisorSigma[ 0,#]]]==1&)],{n,100}] (* _Harvey P. Dale, Jul 02 2022 *)

a(n) = Sum_{k=2..floor(n/2)} [d(k) = d(n-k)] * c(k) * c(n-k), where [ ] is the Iverson bracket, d(n) is the number of divisors of n (A000005), and c is the characteristic function of composite numbers (A066247).

cp's OEIS Frontend

A045917 From Goldbach problem: number of decompositions of 2n into unordered sums of two primes.

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A224712 The number of unordered partitions {a, b} of n such that a or b is composite and the other is prime.

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A224709 The number of unordered partitions {a,b} of the even numbers 2n such that a and b are composite.

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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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A024683 a(n) is the number of ways prime(n) is a sum of two composite numbers r,s satisfying r < s.

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A171691 Number of unordered partitions {k1, k2} of n such that k1 and k2 are nonnegative nonprimes A141468.

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A339927 Number of partitions of n into two composite parts with the same number of divisors.

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