A002375 -id:A002375 - cp's OEIS frontend

A001031 Goldbach conjecture: a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).

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

Offset: 1

N. J. A. Sloane

Number of decompositions of 2*n into sum of two noncomposite numbers. - Omar E. Pol, Dec 12 2024

			1 is counted as a prime, so a(1)=1 since 2=1+1, a(2)=2 since 4=2+2=3+1, ..

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 9.
Deshouillers, J.-M.; te Riele, H. J. J.; and Saouter, Y.; New experimental results concerning the Goldbach conjecture. Algorithmic number theory (Portland, OR, 1998), 204-215, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998.
Apostolos Doxiadis: Uncle Petros and Goldbach's Conjecture, Faber and Faber, 2001
R. K. Guy, Unsolved problems in number theory, second edition, Springer-Verlag, 1994.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 79.
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
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 Math., Vol. 44, No. 1 (1923), pp. 1-70.
Romeo Meštrović, Different classes of binary necklaces and a combinatorial method for their enumerations, arXiv:1804.00992 [math.CO], 2018.
T. Oliveira e Silva, Goldbach conjecture verification
J. Richstein, Verifying the Goldbach conjecture up to 4*10^14, Mathematics of Computation, Vol. 70, No. 236, pp. 1745-1749, 2001.
Matti K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Mathematics of Computation, Vol. 61, No. 204, pp. 931-934, October 1993.
Eric Weisstein's World of Mathematics, Goldbach Partition
Index entries for sequences related to Goldbach conjecture

Cf. A002372 (the main entry), A002373, A002374, A002375, A006307, A008578, A010051, A045917.

a001031 n = sum (map a010051 gs) + fromEnum (1 `elem` gs)
   where gs = map (2 * n -) $ takeWhile (<= n) a008578_list
-- Reinhard Zumkeller, Aug 28 2013

nn = 10^2; ps = Boole[PrimeQ[Range[2*nn]]]; ps[[1]] = 1; Table[Sum[ps[[i]] ps[[2*n - i]], {i, n}], {n, nn}] (* T. D. Noe, Apr 11 2011 *)

a(n)=my(s); forprime(p=2,n, if(isprime(2*n-p), s++)); if(isprime(2*n-1), s+1, s) \\ Charles R Greathouse IV, Feb 06 2017

Not very efficient: a(n) = (Sum_{i=1..n} (pi(i) - pi(i-1))*(pi(2*n-i) - pi(2*n-i-1))) + (pi(2*n-1) - pi(2*n-2)) + floor(1/n). - Wesley Ivan Hurt, Jan 06 2013

a(n) = floor((A096139(n)+1)/2). - Reinhard Zumkeller, Aug 28 2013

More terms from Ray Chandler, Sep 19 2003

A071330 Number of decompositions of n into sum of two prime powers.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 19 2002

a(2*n) > 0 (Goldbach's conjecture).

a(A071331(n)) = 0; A095840(n) = a(A000961(n)).

			10 = 1 + 3^2 = 2 + 2^3 = 3 + 7 = 5 + 5, therefore a(10) = 4;
11 = 2 + 3^2 = 3 + 2^3 = 4 + 7, therefore a(11) = 3;
12 = 1 + 11 = 3 + 3^2 = 2^2 + 2^3 = 5 + 7, therefore a(12) = 4;
a(149)=0, as for all x<149: if x is a prime power then 149-x is not.

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

Cf. A000961, A002375, A010055, A061358, A071331, A109829.

a071330 n = sum $
   map (a010055 . (n -)) $ takeWhile (<= n `div` 2) a000961_list
-- Reinhard Zumkeller, Jan 11 2013

primePowerQ[n_] := Length[ FactorInteger[n]] == 1; a[n_] := (r = 0; Do[ If[ primePowerQ[k] && primePowerQ[n-k], r++], {k, 1, Floor[n/2]}]; r); Table[a[n], {n, 1, 95}](* Jean-François Alcover, Nov 17 2011, after Michael B. Porter *)

ispp(n) = (omega(n)==1 || n==1)
A071330(n) = {local(r);r=0;for(i=1,floor(n/2),if(ispp(i) && ispp(n-i),r++));r} \\ Michael B. Porter, Dec 04 2009

a(n)=my(s); forprime(p=2,n\2,if(isprimepower(n-p), s++)); for(e=2,log(n)\log(2), forprime(p=2, sqrtnint(n\2,e), if(isprimepower(n-p^e), s++))); s+(!!isprimepower(n-1))+(n==2) \\ Charles R Greathouse IV, Nov 21 2014

A099302 Number of integer solutions to x' = n, where x' is the arithmetic derivative of x.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 3, 0, 2, 1, 2, 2, 3, 0, 4, 1, 3, 1, 2, 0, 3, 2, 4, 1, 4, 0, 4, 0, 2, 2, 3, 1, 4, 1, 4, 2, 4, 0, 6, 1, 4, 1, 3, 0, 5, 2, 4, 0, 4, 1, 7, 2, 3, 1, 5, 0, 6, 0, 3, 1, 5, 2, 7, 1, 5, 3, 5, 1, 7, 0, 6, 2, 5, 0, 8, 1, 5, 2, 4, 0, 9, 3, 6, 0, 5, 1, 8, 0, 3, 1, 6, 2, 8, 2, 5, 1, 6

Offset: 2

T. D. Noe, Oct 12 2004, Apr 24 2011

This is the i(n) function in the paper by Ufnarovski and Ahlander. Note that a(1) is infinite because all primes satisfy x' = 1. The plot shows the great difference in the number of solutions for even and odd n. Also compare sequence A189558, which gives the least number have n solutions, and A189560, which gives the least such odd number.

It appears that there are a finite number of even numbers having a given number of solutions. This conjecture is explored in A189561 and A189562.

See A003415

Antti Karttunen, Table of n, a(n) for n = 2..100000 (terms 2..40000 from T. D. Noe)
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A002620, A003415 (arithmetic derivative of n), A099303 (greatest x such that x' = n), A098699 (least x such that x' = n), A098700 (n such that x' = n has no integer solution), A239433 (n such that x' = n has at least one solution).

Cf. A002375 (a lower bound for even n), A369054 (a lower bound for n of the form 4m+3).

a099302 n = length $ filter (== n) $ map a003415 [1 .. a002620 n]
-- Reinhard Zumkeller, Mar 18 2014

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; Table[Count[d1, n], {n, 2, 400}]

up_to = 100000; \\ A002620(10^5) = 2500000000
A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A099302list(up_to) = { my(d,c,v=vector(up_to)); for(i=1, A002620(up_to), d = A003415(i); if(d>1 && d<=up_to, v[d]++)); (v); };
v099302 = A099302list(up_to);
A099302(n) = v099302[n]; \\ Antti Karttunen, Jan 21 2024

from sympy import factorint
def A099302(n): return sum(1 for m in range(1,(n**2>>2)+1) if sum((m*e//p for p,e in factorint(m).items())) == n) # Chai Wah Wu, Sep 12 2022

a(A098700(n)) = 0; a(A239433(n)) > 0. - Reinhard Zumkeller, Mar 18 2014
From Antti Karttunen, Jan 21 2024: (Start)
a(n) = Sum_{i=1..A002620(n)} [A003415(i)==n], where [ ] is the Iverson bracket.
a(2n) >= A002375(n), a(2n+1) >= A369054(2n+1).
(End)

A047160 For n >= 2, a(n) = smallest number m >= 0 such that n-m and n+m are both primes, or -1 if no such m exists.

Original entry on oeis.org

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

Offset: 2

Lior Manor

I have confirmed there are no -1 entries through integers to 4.29*10^9 using PARI. - Bill McEachen, Jul 07 2008
From Daniel Forgues, Jul 02 2009: (Start)
Goldbach's Conjecture: for all n >= 2, there are primes (distinct or not) p and q s.t. p+q = 2n. The primes p and q must be equidistant (distance m >= 0) from n: p = n-m and q = n+m, hence p+q = (n-m)+(n+m) = 2n.
Equivalent to Goldbach's Conjecture: for all n >= 2, there are primes p and q equidistant (distance >= 0) from n, where p and q are n when n is prime.
If this conjecture is true, then a(n) will never be set to -1.
Twin Primes Conjecture: there is an infinity of twin primes.
If this conjecture is true, then a(n) will be 1 infinitely often (for which each twin primes pair is (n-1, n+1)).
Since there is an infinity of primes, a(n) = 0 infinitely often (for which n is prime).
(End)
If n is composite, then n and a(n) are coprime, because otherwise n + a(n) would be composite. - Jason Kimberley, Sep 03 2011
From Jianglin Luo, Sep 22 2023: (Start)
a(n) < primepi(n)+sigma(n,0);
a(n) < primepi(primepi(n)+n);
a(n) < primepi(n), for n>344;
a(n) = o(primepi(n)), as n->+oo. (End)
If -1 < a(n) < n-3, then a(n) is divisible by 3 if and only if n is not divisible by 3, and odd if and only if n is even. - Robert Israel, Oct 05 2023

			16-3=13 and 16+3=19 are primes, so a(16)=3.

T. D. Noe, Table of n, a(n) for n = 2..10000
Jason Kimberley, Symmetrical plot of A047160

Cf. A001031, A002092, A002372, A002373, A002374, A002375, A014092, A025583, A035026, A047949, A071406, A082467, A102084, A103147, A112823, A155764, A155765, A177461, A078611, A010051, A045917, A325142.

a047160 n = if null ms then -1 else head ms
            where ms = [m | m <- [0 .. n - 1],
                            a010051' (n - m) == 1, a010051' (n + m) == 1]
-- Reinhard Zumkeller, Aug 10 2014

A047160:=func;[A047160(n):n in[2..100]]; // Jason Kimberley, Sep 02 2011

Table[k = 0; While[k < n && (! PrimeQ[n - k] || ! PrimeQ[n + k]), k++]; If[k == n, -1, k], {n, 2, 100}]
smm[n_]:=Module[{m=0},While[AnyTrue[n+{m,-m},CompositeQ],m++];m]; Array[smm,100,2] (* Harvey P. Dale, Nov 16 2024 *)

a(n)=forprime(p=n,2*n, if(isprime(2*n-p), return(p-n))); -1 \\ Charles R Greathouse IV, Jun 23 2017

10 N=2// 20 M=0// 30 if and{prmdiv(N-M)=N-M,prmdiv(N+M)=N+M} then print M;:goto 50// 40 inc M:goto 30// 50 inc N: if N>130 then stop// 60 goto 20

a(n) = n - A112823(n).

a(n) = A082467(n) * A005171(n), for n > 3. - Jason Kimberley, Jun 25 2012

More terms from Patrick De Geest, May 15 1999
Deleted a comment. - T. D. Noe, Jan 22 2009
Comment corrected and definition edited by Daniel Forgues, Jul 08 2009

A001172 Smallest even number that is an unordered sum of two odd primes in exactly n ways.

Original entry on oeis.org

0, 6, 10, 22, 34, 48, 60, 78, 84, 90, 114, 144, 120, 168, 180, 234, 246, 288, 240, 210, 324, 300, 360, 474, 330, 528, 576, 390, 462, 480, 420, 570, 510, 672, 792, 756, 876, 714, 798, 690, 1038, 630, 1008, 930, 780, 960, 870, 924, 900, 1134, 1434, 840, 990, 1302

Offset: 0

N. J. A. Sloane, Eric Wolman, Dec 17 1969

a(n) = A023036(n) for all n > 1.

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).

T. D. Noe, Table of n, a(n) for n = 0..10000
Eric Wolman, Letter to N. J. A. Sloane, 1970.
Index entries for sequences related to Goldbach conjecture

Cf. A000954, A000974, A002375, A005843, A023036.

Divided by 2: A258713.

nn = 55; a = Table[0, {nn}]; n = 6; While[Times @@ a == 0, c = 0; k = 3; While[k <= n/2, If[PrimeQ[k] && PrimeQ[n - k], c++]; k++]; If[c <= nn && a[[c]] == 0, a[[c]] = n]; n = n + 2]; Prepend[a, 0]

a(0) corrected by Zak Seidov, Sep 30 2011

A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1.

Original entry on oeis.org

9, 161, 209, 221, 2189, 2561, 3281, 3629, 5249, 5549, 6401, 7181, 7661, 8321, 8909, 9089, 9869, 10001, 10349, 10541, 10961, 11009, 11021, 29861, 38981, 52601, 66149, 84101, 93029, 97481, 132809, 150281, 158969, 163301, 197669, 214661, 227321, 235721, 285449, 321989, 338021, 357881, 369701, 381449, 385349, 416261, 420089, 442889

Offset: 1

Antti Karttunen, Oct 09 2019

nonn

Numbers n such that A327859(n) = A276086(A003415(n)) is an odd prime.
Composite terms in A328232.
Although it first might seem that the numbers whose arithmetic derivative is A002110(k) all appear before any of those whose arithmetic derivative is A002110(k+1), that is not true, as for example, we have a(56) = 570149, and A003415(570149) = 2310, a(57) = 570209, and A003415(570209) = 30030, but then a(58) = 573641 with A003415(573641) = 2310 again.
Because this is a subsequence of A327862 (all primorials > 1 are of the form 4k+2), only odd numbers are present.
Conjecture: No multiples of 5 occur in this sequence, and no multiples of 3 after the initial 9.
Of the first 10000 terms, all others are semiprimes (with 9 the only square one), except 1547371 = 7^2 * 23 * 1373 and 79332523 = 17^2 * 277 * 991, the latter being the only known term whose decimal expansion ends with 3. If all solutions were semiprimes p*q such that p+q = A002110(k) for some k > 1 (see A002375), it would be a sufficient reason for the above conjecture to hold. - David A. Corneth and Antti Karttunen, Oct 11 2019
In any case, the solutions have to be of the form "odd numbers with an even number of prime factors with multiplicity" (see A235992), and terms must also be cubefree (A004709), as otherwise the arithmetic derivative would not be squarefree.
Sequence A366890 gives the non-Goldbachian solutions, i.e., numbers that are not semiprimes. See also A368702. - Antti Karttunen, Jan 17 2024

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1078 terms from Antti Karttunen)
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to primorial numbers

Cf. A002110, A002375, A002620, A003415, A024451, A143293, A157037, A276150, A327859, A327969, A328233, A328243.
Cf. A351029 (number of k for which k' = A002110(n)).
Cf. A368703, A368704 (the least and the greatest k for which k' = A002110(n)).
Cf. A366890 (terms that are not semiprimes), A368702 (numbers k such that k' is one of the terms of this sequence).
Subsequence of following sequences: A004709, A189553, A327862, A328232, A328234.

ad[n_] := n * Total @ (Last[#]/First[#] & /@ FactorInteger[n]); primQ[n_] := Max[(f = FactorInteger[n])[[;;,2]]] == 1 && PrimePi[f[[-1,1]]] == Length[f]; Select[Range[10^4], primQ[ad[#]] &] (* Amiram Eldar, Oct 11 2019 *)

A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
isA327978flat(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(u))); }; \\ Slow!
k=0; for(n=1,A002620(30030),if(isA327978flat(n), k++; write("b327978.txt", k, " ", n)));

A327969(a(n)) = 4 for all n.

A129363 Number of partitions of 2n into the sum of two twin primes.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 1, 2, 2, 3, 3, 2, 1, 2, 2, 3, 4, 2, 1, 2, 1, 2, 3, 3, 2, 2, 1, 2, 4, 3, 3, 4, 2, 2, 3, 2, 2, 4, 2, 0, 0, 0, 2, 4, 3, 2, 2, 2, 4, 6, 3, 3, 5, 3, 1, 2, 1, 2, 4, 2, 1, 2, 2, 4, 5, 3, 2, 4, 3, 3, 4, 2, 2, 4, 2, 3, 6, 3, 1, 2, 1, 3, 6, 4, 2, 2, 1, 2, 4, 3, 4, 6, 4, 4, 5, 3, 6, 12

Offset: 1

T. D. Noe, Apr 11 2007

nonn

a(n/2)=0 for the n in A007534. The logarithmic plot of this sequence seems very regular after 200000 terms.

			a(11)=3 because 22 = 3+19 = 5+17 = 11+11.

T. D. Noe, Table of n, a(n) for n=1..10000
James Grime and Brady Haran, Goldbach Conjecture (but with TWIN PRIMES), Numberphile video (2024)
T. D. Noe, Logarithmic plot of 10^6 terms

Cf. A175931 (n for which a(n-1), a(n), a(n+1) are equal).

Cf. A001097, A002375, A007534.

a129363 n = sum $ map (a164292 . (2*n -)) $ takeWhile (<= n) a001097_list
-- Reinhard Zumkeller, Feb 03 2014

nn=1000; tw=Select[Prime[Range[PrimePi[nn]]], PrimeQ[ #+2]&]; tw=Union[tw,tw+2]; tc=Table[0,{nn}]; tc[[tw]]=1; Table[cnt=0; k=1; While[tw[[k]]<=n/2, cnt=cnt+tc[[n-tw[[k]]]]; k++ ]; cnt, {n,2,nn,2}]

a(n) = Sum_{i=1..n} ceiling((A010051(i+2) + A010051(i-2))/2) * ceiling((A010051(2n-i+2) + A010051(2n-i-2))/2) * A010051(2n-i) * A010051(i). - Wesley Ivan Hurt, Jan 30 2014

a(n) = sum(A164292(2*n - A001097(k)): A001097(k) <= n). - Reinhard Zumkeller, Feb 03 2014

Comment converted to crossref by Klaus Brockhaus, Oct 27 2010

A219055 Number of ways to write n = p+q(3-(-1)^n)/2 with p>q and p, q, p-6, q+6 all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 1, 3, 1, 0, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 1, 3, 2, 1, 4, 1, 0, 3, 3, 1, 3, 1, 1, 3, 3, 1, 2, 2, 2, 2, 2, 2, 3, 1, 3, 3, 1, 2, 6, 1, 2, 2, 1, 3, 5, 0, 1, 4, 2, 1, 4, 0, 1, 4, 3

Offset: 1

Zhi-Wei Sun, Nov 11 2012

Conjecture: a(n) > 0 for all even n > 8012 and odd n > 15727.
This implies Goldbach's conjecture, Lemoine's conjecture and the conjecture that there are infinitely many primes p with p+6 also prime.
It has been verified for n up to 10^8.
Zhi-Wei Sun also made the following general conjecture: For any two multiples d_1 and d_2 of 6, all sufficiently large integers n can be written as p+q(3-(-1)^n)/2 with p>q and p, q, p-d_1, q+d_2 all prime. For example, for (d_1,d_2) = (-6,6),(-6,-6),(6,-6),(12,6),(-12,-6), it suffices to require that n is greater than 15721, 15733, 15739, 16349, 16349 respectively.

			a(18) = 2 since 18 = 5+13 = 7+11 with 5+6, 13-6, 7+6, 11-6 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A023201, A002375, A046927, A218754, A218585, A218654, A218825, A219023, A219026, A219052.

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]-6]==True,1,0],{k,1,PrimePi[(n-1)/(2+Mod[n,2])]}]
Do[Print[n," ",a[n]],{n,1,100000}]

A219055(n)={my(c=1+bittest(n, 0), s=0); forprime(q=1, (n-1)\(c+1), isprime(q+6) && isprime(n-c*q) && isprime(n-c*q-6) && s++); s} \\ M. F. Hasler, Nov 11 2012

A230219 Number of ways to write 2*n + 1 = p + q + r with p <= q such that p, q, r are primes in A230217 and p + q + 9 is also prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 5, 2, 2, 4, 3, 1, 4, 3, 1, 4, 1, 2, 5, 2, 3, 4, 3, 3, 8, 6, 3, 12, 6, 2, 13, 3, 3, 7, 6, 4, 5, 4, 4, 8, 7, 4, 12, 7, 3, 19, 6, 3, 16, 5, 4, 9, 5, 5, 7, 10, 4, 5, 8, 3, 14, 4, 3, 14, 2, 5, 12, 5, 2, 14, 9, 2, 10, 12, 4, 12, 7, 6, 12, 7, 9, 14, 8, 6, 12, 5, 4, 19, 8, 4, 23, 6, 3, 14

Offset: 1

Zhi-Wei Sun, Oct 11 2013

nonn

Conjecture: a(n) > 0 for all n > 6.
This implies Goldbach's weak conjecture for odd numbers and also Goldbach's conjecture for even numbers.
The conjecture also implies that there are infinitely many primes in A230217.

			a(18) = 1 since 2*18 + 1 = 7 + 13 + 17, and 7, 13, 17 are terms of A230217, and 7 + 13 + 9 = 29 is a prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A002375, A068307, A230217, A230140, A230141.

RQ[n_]:=PrimeQ[n+6]&&PrimeQ[3n+8]
SQ[n_]:=PrimeQ[n]&&RQ[n]
a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&PrimeQ[Prime[i]+Prime[j]+9]&&SQ[2n+1-Prime[i]-Prime[j]],1,0],{i,1,PrimePi[n-1]},{j,i,PrimePi[2n-2-Prime[i]]}]
Table[a[n],{n,1,100}]

A278700 Number of Goldbach partitions (p,q) of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 1, 0, 2, 0, 1, 3, 1, 0, 3, 0, 1, 2, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 5, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 3, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 1

Wesley Ivan Hurt, Nov 26 2016

Records are a(3) = 1, a(5) = 2, a(12) = 3, a(30) = 5, a(165) = 6, a(8021811) = 7. - Charles R Greathouse IV, Nov 30 2016

a(n) <= A002375(n). - Wesley Ivan Hurt, Dec 17 2016

			a(5) = 2; There are 2 Goldbach partitions of 2*5 = 10, namely (3,7) and (5,5). (3,7) satisfies the condition that every prime from 3 to 7 (inclusive) appears as a part in some Goldbach partition of 10. This is also true of (5,5) since 5 appears in its own partition.

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A000720, A002375, A010051, A055381, A279103.

Table[Sum[Times @@ Map[Boole@ PrimeQ@ # &, {i, 2 n - i}] * Product[(1 - Abs[Subtract @@ Map[Boole@ PrimeQ@ # &, {k, 2 n - k}]]), {k, i, n}], {i, 3, n}], {n, 89}] (* Michael De Vlieger, Nov 30 2016 *)

a(n) = sum(i=3, n, (ispseudoprime(i) * ispseudoprime(2*n-i) * prod(k=i, n, (1-abs(ispseudoprime(k)-ispseudoprime(2*n-k)))))) \\ Felix Fröhlich, Nov 28 2016

a(n) = if(n<3, return(0)); my(s,p=n,N=2*n); forprime(q=n, N, while(pCharles R Greathouse IV, Nov 30 2016

a(n) = Sum_{i=3..n} (c(i) * c(2*n-i) * Product_{k=i..n} (1 - abs(c(k) - c(2*n-k)))), where c is the prime characteristic (A010051).

cp's OEIS Frontend

A001031 Goldbach conjecture: a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A071330 Number of decompositions of n into sum of two prime powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

A099302 Number of integer solutions to x' = n, where x' is the arithmetic derivative of x.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A047160 For n >= 2, a(n) = smallest number m >= 0 such that n-m and n+m are both primes, or -1 if no such m exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

UBASIC

Formula

Extensions

A001172 Smallest even number that is an unordered sum of two odd primes in exactly n ways.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI