A061357 -id:A061357 - cp's OEIS frontend

A002372 Goldbach conjecture: number of decompositions of 2n into ordered sums of two odd primes.

0, 0, 1, 2, 3, 2, 3, 4, 4, 4, 5, 6, 5, 4, 6, 4, 7, 8, 3, 6, 8, 6, 7, 10, 8, 6, 10, 6, 7, 12, 5, 10, 12, 4, 10, 12, 9, 10, 14, 8, 9, 16, 9, 8, 18, 8, 9, 14, 6, 12, 16, 10, 11, 16, 12, 14, 20, 12, 11, 24, 7, 10, 20, 6, 14, 18, 11, 10, 16, 14, 15, 22, 11, 10, 24, 8, 16, 22, 9, 16, 20, 10

Offset: 1

N. J. A. Sloane

The weak form of this conjecture was proved by Helfgott (see link below). - T. D. Noe, May 14 2013
Goldbach conjectured in 1742 that for n >= 3, this sequence never vanishes. This is still unproved.
Number of different primes occurring when 2n is expressed as p1+q1 = ... = pk+qk where pk,qk are odd primes with pk <= qk. For example when n=5: 10 = 3+7 = 5+5, we can see 3 different primes so a(5) = 3. - Naohiro Nomoto, Feb 24 2002
Comments from Tomás Oliveira e Silva to Number Theory List, Feb 05 2005: With the help of Siegfied "Zig" Herzog of PSU, I was able to verify the Goldbach conjecture up to 2e17. Let 2n=p+q, with p and q prime be a Goldbach partition of 2n. In a minimal Goldbach partition p is as small as possible. The largest p of a minimal Goldbach partition found was 8443 and is needed for 2n=121005022304007026. Furthermore, the largest prime gap found was 1220-1; it occurs after the prime 80873624627234849.
Comments from Tomás Oliveira e Silva to Number Theory List, Apr 26 2007: With the help of Siegfried "Zig" Herzog, the NCSA and others, I have just finished the verification of the Goldbach conjecture up to 1e18. This took about 320 years of CPU time, including a double-check of the results up to 1e17. As expected, no counterexample to the conjecture was found. As side results, the number of twin primes up to 1e18 was also computed, as was the number of primes in each of the residue classes modulo 120. Also, the number of occurrences of each (observed) prime gap was also recorded.
For n > 2 we have a(n) = 2*A002375(n)-1 if n is prime and a(n) = 2*A002375(n) if n is composite. - Emeric Deutsch, Jul 14 2004
For n > 2, a(n) = 2*A002375(n) - A010051(n). - Jason Kimberley, Aug 31 2011
a(n) = Sum_{p odd prime < 2*n} A010051(2*n - p). - Reinhard Zumkeller, Oct 19 2011
There is an interesting similarity with square numbers: The number of divisors of n is odd iff n is square (A000290).  The number of decompositions of 2n into ordered sums of two primes (equaling the number of the unique primes in all such decompositions) is odd iff n is prime. - Ivan N. Ianakiev, Feb 28 2015

			2 has no such decompositions, so a(1) = 0.
Idem for 4, whence a(2) = 0.
6 = 3+3, so a(3) = 1.
8 = 3+5 = 5+3, so a(4) = 2.
10 = 5+5 = 3+7 = 7+3, so a(5) = 3.
12 = 5+7 = 7+5; so a(6) = 2, etc.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 9.
R. K. Guy, Unsolved problems in number theory, second edition, Springer-Verlag, 1994.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 2.8 (for Goldbach conjecture).
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 79, 80.
N. Pipping, Neue Tafeln für das Goldbachsche Gesetz nebst Berichtigungen zu den Haussnerschen Tafeln, Finska Vetenskaps-Societeten, Comment. Physico Math. 4 (No. 4, 1927), pp. 1-27.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. L. Stein and P. R. Stein, Tables of the Number of Binary Decompositions of All Even Numbers Less Than 200,000 into Prime Numbers and Lucky Numbers. Report LA-3106, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Sep 1964.

T. D. Noe, Table of n, a(n) for n = 1..10000
Peter B. Borwein, Stephen K. K. Choi, Greg Martin, Charles L. Samuels, Polynomials whose reducibility is related to the Goldbach conjecture, arXiv:1408.4881 [math.NT], 2014 (see R(N) on page 1).
J. -M. Deshouillers, H. J. J. te Riele, Y. Saouter, New experimental results concerning the Goldbach conjecture, preprint, Centrum Wiskunde & Informatica, 1998.
J. -M. Deshouillers, H. J. J. te Riele, Y. Saouter, New experimental results concerning the Goldbach conjecture, Algorithmic number theory (Portland, OR, 1998), 204-215, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998.
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1922.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013-2014.
Yan Kun, Li Hou Biao, Divisor Goldbach Conjecture and its Partition Number, arXiv:1603.05233 [math.NT], 2016.
T. Oliveira e Silva, Goldbach conjecture verification.
T. Oliveira e Silva, Gaps between consecutive primes.
T. Oliveira e Silva, Tables of values of pi(x) and of pi2(x).
T. Oliveira e Silva, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060. - _Felix Fröhlich_, Jun 23 2014
Jörg Richstein, Verifying the Goldbach conjecture up to 4 * 10^14, Math. Comput., 70 (2001), 1745-1749.
Matti K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Math. Comp. 61 (1993), pp. 931-934.
Eric Weisstein's World of Mathematics, Goldbach Conjecture.
A. Zaccagnini, Goldbach Variations: problems with prime numbers.
Index entries for sequences related to Goldbach conjecture

Essentially identical to A035026.
Cf. A002375 (unordered sums), A002374, A014092, A035026, A059998, A001031, A002373, A045917, A006307.
Cf. A065091, A010051.
Cf. A069360, A085090.

a002372 n = sum $ map (a010051 . (2*n -)) $ takeWhile (< 2*n) a065091_list
-- Reinhard Zumkeller, Oct 19 2011

A002372 := func; [A002372(n):n in[1..82]]; // Jason Kimberley, Sep 01 2011

a:=proc(n) local c,k; c:=0: for k from 1 to n do if isprime(2*k+1)=true and isprime(2*n-2*k-1)=true then c:=c+1 else c:=c fi od end: seq(a(n),n=1..82); # Emeric Deutsch, Jul 14 2004

For[lst={}; n=1, n<=100, n++, For[cnt=0; i=1, i<=2n-1, i++ If[OddQ[i]&&PrimeQ[i]&&PrimeQ[2n-i], cnt++ ]]; AppendTo[lst, cnt]]; lst
(* second program: *)
A002372[n_] := Module[{i = 0}, Do[If[PrimeQ[2 n - Prime@p], i++], {p, 2, PrimePi[2 n - 3]}]; i]; Array[A002372, 82] (* JungHwan Min, Aug 24 2016 *)
i[n_] := If[PrimeQ[2 n - 1], 2 n - 1, 0]; A085090 = Array[i, 82];
r[n_] := Table[A085090[[k]] + A085090[[n - k + 1]], {k, 1, n}];
countzeros[l_List] := Sum[KroneckerDelta[0, k], {k, l}];
Table[n - 2 countzeros[A085090[[1 ;; n]]] + countzeros[r[n]],
{n, 1, 82}] (* Fred Daniel Kline, Aug 13 2018 *)
countPrimes[n_] := Sum[KroneckerDelta[True, PrimeQ[2 m - 1],
PrimeQ[2 (n - m + 1) - 1]], {m, 1, n}]; Array[countPrimes, 82] (* Fred Daniel Kline, Oct 07 2018 *)

isop(n) = (n % 2) && isprime(n);
a(n) = n*=2; sum(i=1, n-1, isop(i)*isop(n-i)); \\ Michel Marcus, Aug 22 2014 and May 28 2020

from sympy import isprime, primerange
def a(n): return sum([1 for p in primerange(3, 2*n-2) if isprime(2*n-p)])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 23 2017

a(n) = A010051(n) + 2*A061357(n), n > 2. - R. J. Mathar, Aug 19 2013

More terms from Larry Reeves (larryr(AT)acm.org), Jun 13 2002

Edited by M. F. Hasler, May 03 2019

A035026 Number of times that i and 2n-i are both prime, for i = 1, ..., 2n-1.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 4, 4, 4, 5, 6, 5, 4, 6, 4, 7, 8, 3, 6, 8, 6, 7, 10, 8, 6, 10, 6, 7, 12, 5, 10, 12, 4, 10, 12, 9, 10, 14, 8, 9, 16, 9, 8, 18, 8, 9, 14, 6, 12, 16, 10, 11, 16, 12, 14, 20, 12, 11, 24, 7, 10, 20, 6, 14, 18, 11, 10, 16, 14, 15, 22, 11, 10, 24, 8, 16, 22, 9, 16, 20, 10

Offset: 1

Gordon R. Bower (siegmund(AT)mosquitonet.com)

a(n) is the convolution of terms 1 to 2n of the characteristic function of the primes, A010051, with itself. Related to Goldbach's conjecture that every even number can be expressed as the sum of two primes. - T. D. Noe, Aug 01 2002
The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002
Total number of printer jobs in all possible schedules for n time slots in the first-come-first-served (FCFS) policy.
a(n) = Sum_{p prime < 2*n} A010051(2*n - p). - Reinhard Zumkeller, Oct 19 2011
For n > 1: length of n-th row of triangle A171637. - Reinhard Zumkeller, Mar 03 2014
a(n) = A001221(A238711(n)) = A238778(n) / n. - Reinhard Zumkeller, Mar 06 2014
From Robert G. Wilson v, Dec 15 2016: (Start)
First occurrence of k: 1, 2, 4, 5, 8, 11, 12, 17, 18, 37, 24, 53, 30, 89, 39, 71, 42, 101, 45, 179, 57, 137, 72, 193, 60, 233, ..., .
Conjectured last occurrence of k: 1, 3, 6, 19, 34, 31, 64, 61, 76, 79, 94, 83, 166, 199, 136, 181, 184, 229, 244, 271, 316, 277, 346, 313, 301, 293, ..., .
Conjectured number occurrences of k: 1, 2, 2, 3, 6, 3, 8, 4, 7, 5, 11, 5, 11, 8, 10, 3, 17, 7, 16, 3, 13, 8, 21, 4, 12, 3, 22, 7, 20, 8, 15, ..., .
Records: 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 24, 26, 28, 38, 42, 48, 54, 60, 64, 82, 88, 102, 104, 114, 116, 136, 146, 152, 166, 182, ..., .
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 500000 terms
Index entries for sequences related to Goldbach conjecture

Cf. A010051. Essentially the same as A002372.

Cf. A073610.

a035026 n = sum $ map (a010051 . (2 * n -)) $
   takeWhile (< 2 * n) a000040_list
-- Reinhard Zumkeller, Oct 19 2011

A035026 := proc(n)
    local a,i ;
    a := 0 ;
    for i from 1 to 2*n-1 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

For[lst={}; n=1, n<=100, n++, For[cnt=0; i=1, i<=2n-1, i++ If[PrimeQ[i]&&PrimeQ[2n-i], cnt++ ]]; AppendTo[lst, cnt]]; lst
f[n_] := Block[{c = Boole@ PrimeQ[ n/2], p = 2}, While[ 2p < n, If[ PrimeQ[n - p], c += 2]; p = NextPrime@ p]; c];; Array[ f[ 2#] &, 90] (* Robert G. Wilson v, Dec 15 2016 *)

For n > 1, a(n) = 2*A045917(n) - A010051(n).
a(n) = A010051(n) + 2*A061357(n). - Wesley Ivan Hurt, Aug 21 2013
a(n) = A073610(2*n). - Ridouane Oudra, Sep 06 2023

Corrected by T. D. Noe, May 05 2002

A092953 Number of primes of the form n+p, where p is a prime < n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 0, 2, 1, 2, 1, 3, 0, 2, 1, 3, 1, 3, 0, 3, 1, 2, 0, 6, 0, 4, 1, 3, 1, 6, 0, 3, 0, 4, 1, 6, 0, 4, 1, 5, 1, 8, 0, 4, 1, 4, 0, 7, 0, 6, 1, 4, 0, 9, 0, 8, 1, 4, 1, 11, 0, 5, 0, 5, 1, 11, 0, 6, 1, 8, 1, 9, 0, 4, 0, 7, 1, 11, 0, 7, 1, 4, 0, 13, 0, 7, 1, 5, 0, 15, 0, 7, 0, 8, 1, 13, 0, 8, 1, 9, 1, 11

Offset: 1

Amarnath Murthy, Mar 24 2004

nonn

Might be called the additive primability of n.

a(A007921(n))=0; for n > 2: a(A030173(n)) > 0 and a(A040976(n)) = 1. - Reinhard Zumkeller, Nov 10 2012

			a(26) = 4: the primes are 29, 31, 37 and 43.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A092954.

Cf. A061357.

a092953 n = sum $
   zipWith (\u v -> a010051' u * a010051' v) [1 .. n - 1] [n + 1 ..]
-- Reinhard Zumkeller, Nov 10 2012

for(n=1,105,c=0;forprime(p=2,n-1,if(isprime(n+p),c++));print1(c,","))

More terms from Klaus Brockhaus and Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 25 2004

A154786 Row sums of triangle in A154725.

Original entry on oeis.org

0, 0, 0, 8, 10, 12, 14, 32, 36, 40, 44, 72, 52, 56, 90, 64, 102, 144, 38, 120, 168, 132, 138, 240, 200, 156, 270, 168, 174, 360, 124, 320, 396, 136, 350, 432, 296, 380, 546, 320, 328, 672, 344, 352, 810, 368, 376, 672, 294, 600, 816, 520, 530, 864, 660, 784, 1140

Offset: 1

Omar E. Pol, Jan 15 2009, Jan 16 2009

Cf. A154724, A154725, A154726, A154727, A154783, A154784, A154785.

Cf. A005843, A061357, A154720, A154787. - Omar E. Pol, Jan 20 2009

A154786 := proc(n) local a,d; a := 0 ; for d from 1 to n-2 do if isprime(n-d) and isprime(n+d) then a := a+2*n; fi; od: a ; end: for n from 1 to 80 do printf("%d,",A154786(n)) ; od: # R. J. Mathar, Jan 18 2009

a(n) = A154785(n) - n.

a(n) = A005843(n)*A061357(n). - Omar E. Pol, Jan 20 2009

Edited by Omar E. Pol, Jan 17 2009

Extended by R. J. Mathar, Jan 18 2009

A154804 Number of ways to represent 2*n as the sum of two distinct primes (counting 1 as a prime).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 3, 3, 2, 4, 2, 2, 4, 3, 3, 4, 2, 3, 5, 4, 3, 6, 4, 3, 6, 3, 3, 7, 3, 5, 6, 3, 5, 7, 5, 5, 7, 5, 4, 9, 4, 4, 10, 4, 4, 7, 4, 6, 9, 6, 5, 9, 7, 7, 11, 6, 5, 12, 3, 5, 10, 4, 7, 10, 5, 5, 9, 8, 7, 11, 5, 5, 13, 5, 8, 11, 5, 8, 10, 6, 5, 14, 9, 6, 12, 7, 6, 15, 7, 8, 13, 5, 8, 12, 7, 9

Offset: 1

Omar E. Pol, Jan 16 2009

nonn

Number of ways to represent 2*n as the sum of two distinct noncomposite numbers. - Omar E. Pol, Dec 11 2024

Cf. A008578, A154720, A154721, A154722, A154723.

a(n) = A101264(n-1) + A061357(n). [From R. J. Mathar, Jan 21 2009]

a(n) = A001031(n) - A080339(n).

More terms from R. J. Mathar, Jan 21 2009

Edited by Franklin T. Adams-Watters, Jan 31 2009

A236747 Number of 0 <= k <= sqrt(n) such that n-k and n+k are both prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 0, 2, 1, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 0

Offset: 1

Juri-Stepan Gerasimov, Jan 30 2014

nonn

Probably a(n) > N for any N and all sufficiently large n. Perhaps a(2591107) is the last 0 in this sequence. - Charles R Greathouse IV, Jan 30 2014

Primes p such that a(p)=1: 2, 3, 7, 11, 13, 17, 19, ... . Juri-Stepan Gerasimov, Feb 02 2014

			a(3) = 1 because 3 - 0 = 3 and 3 + 0 = 3 are both prime for k = 0;
a(4) = 1 because 4 - 1 = 3 and 4 + 1 = 5 are both prime for k = 1 < sqrt(4) = 2;
a(5) = 2 because 5 - 0 = 5 and 5 + 0 = 5 are both prime for k = 0, 5 - 2 = 3 and 5 + 2 = 7 are both prime for k = 2 < sqrt(5).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000196, A010051, A061357, A171637.

A236767 := proc(n)
    local a,k ;
    a := 0 ;
    for k from 0 to floor(sqrt(n)) do
        if isprime(n-k) and isprime(n+k) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A236767(n),n=1..80) ; # R. J. Mathar, Dec 01 2020

Table[Length[Select[Range[0, Sqrt[n]], PrimeQ[n - #] && PrimeQ[n + #] &]], {n, 100}] (* T. D. Noe, Feb 01 2014 *)

a(n)=sum(k=0,sqrtint(n),isprime(n-k)&&isprime(n+k)) \\ Charles R Greathouse IV, Jan 30 2014

(define (A236747 n) (add (lambda (k) (* (A010051 (- n k)) (A010051 (+ n k)))) 0 (A000196 n)))
;; The following implements sum_{i=lowlim..uplim} intfun(i):
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
;; From Antti Karttunen, Feb 01 2014

a(n) = Sum_{k=0..A000196(n)} (A010051(n-k) * A010051(n+k)). - Antti Karttunen, Feb 01 2014

Terms recomputed (with corrections) by Antti Karttunen, Feb 01 2014

A072511 Least number m such that 2m can be expressed as the sum of two distinct primes in exactly n ways.

Original entry on oeis.org

1, 4, 8, 12, 18, 24, 30, 39, 42, 45, 57, 72, 60, 84, 90, 117, 123, 144, 120, 105, 162, 150, 180, 237, 165, 264, 288, 195, 231, 240, 210, 285, 255, 336, 396, 378, 438, 357, 399, 345, 519, 315, 504, 465, 390, 480, 435, 462, 450, 567, 717, 420, 495, 651, 540, 615

Offset: 0

Amarnath Murthy, Jul 24 2002

nonn

Let f(x) = A061357(x) be the number of primes p < x such that 2x-p is also prime. a(n) is the smallest positive integer x such that f(x) = n.
Or, least number m such that m can be expressed as the mean of two distinct primes in exactly n ways. Cf. A061357 = number of ways n can be expressed as the mean of two distinct primes, A061357 = number of ways the even integer 2n can be written as the sum of two primes for all even integers >6. - Zak Seidov, Sep 08 2006
For what values of n is a(n) > a(n+1)?

			a(1)=4 because 8 = 3+5 that is 8 can be expressed as the sum of two distinct primes by exactly 1 way,
a(2)=8 because 16 = 3+13 = 5+11 (2 ways),
a(3)=12 because 24 = 5+17 = 7+17 = 11+17 (3 ways),
a(4)=18 because 36 = 5+31 = 7+29 = 13+23 = 17+19 (4 ways), etc.
Starting with third term 12, all terms are multiples of 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A061357.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a072511 = (+ 1) . fromJust . (`elemIndex` a061357_list)
-- Reinhard Zumkeller, Nov 10 2012

f[x_] := Length[Select[2x-(Prime/@Range[PrimePi[x-1]]), PrimeQ]]; For[x=1, x<1000, x++, fx=f[x]; If[a[fx]>=0, Null, Null, a[fx]=x]]; a/@Range[0, 60]

It seems that for n>7 n*log(n)*log(log(n)) < a(n) < 3n*log(n)*log(log(n)). Does lim n->infinity a(n)/n/log(n)/log(log(n)) exist ? - Benoit Cloitre, Aug 11 2002

Edited by Dean Hickerson, Aug 07 2002

Entry revised by N. J. A. Sloane, Sep 12 2006

cp's OEIS Frontend

A154787 a(n) = A061357(n)*n = A154786(n)/2.

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A002372 Goldbach conjecture: number of decompositions of 2n into ordered sums of two odd primes.

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A035026 Number of times that i and 2n-i are both prime, for i = 1, ..., 2n-1.

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A092953 Number of primes of the form n+p, where p is a prime < n.

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A154786 Row sums of triangle in A154725.

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A154804 Number of ways to represent 2*n as the sum of two distinct primes (counting 1 as a prime).

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A236747 Number of 0 <= k <= sqrt(n) such that n-k and n+k are both prime.

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A356864 a(n) is the number of primes p < n such that 2n-p and p(2n-p)+2n are also prime.