A002373 -id:A002373 - cp's OEIS frontend

A273457 Even numbers 2n that do not have a Goldbach partition 2n = p + q (p < q; p, q prime) satisfying sqrt(n) < p <= sqrt(2n).

2, 4, 6, 8, 12, 18, 20, 22, 24, 26, 30, 32, 38, 40, 44, 52, 56, 58, 62, 64, 70, 72, 76, 82, 84, 88, 92, 94, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 126, 128, 130, 132, 134, 136, 140, 144, 146, 152, 154, 156, 158, 164, 166, 172, 182, 188, 196, 198, 200, 214

Offset: 1

Corinna Regina Böger, Dec 11 2016

nonn

This is an extension of A244408.
There are 74 elements of A279040 that are also in this sequence. These common elements are in A244408.
It is conjectured that a(12831) = 15702604 is the last term. There are no more terms below 4*10^10.

			32 is in the sequence because 32 has two Goldbach partitions: 32 = 3 + 29 with 3 < sqrt(16) and 32 = 13 + 19 with 13 > sqrt(32).

Corinna Regina Böger, Table of n, a(n) for n = 1..12831

Cf. A002373, A020481, A025018, A025019, A244408, A279040.

noGoldbatSqrQ[n_] := Block[{p = NextPrime[Sqrt[n/2]]}, While[2p < n && !PrimeQ[n - p], p = NextPrime@ p]; p > Sqrt[n]]; noGoldbatSqrQ[4] = True; Select[2Range[107], noGoldbatSqrQ] (* Robert G. Wilson v, Dec 15 2016 *)

noSpecialGoldbach(n) = forprime(p=sqrtint(n/2-1) + 1, sqrtint(n), if(p<(n-p) && isprime(n-p), return(0))); 1
is(n) = n%2 == 0 && noSpecialGoldbach(n)

A305883 Triangle read by rows: row n lists the pairs (p, q) such that p, q are primes, p+q=2*n and p < q.

Original entry on oeis.org

3, 5, 3, 7, 5, 7, 3, 11, 3, 13, 5, 11, 5, 13, 7, 11, 3, 17, 7, 13, 3, 19, 5, 17, 5, 19, 7, 17, 11, 13, 3, 23, 7, 19, 5, 23, 11, 17, 7, 23, 11, 19, 13, 17, 3, 29, 13, 19, 3, 31, 5, 29, 11, 23, 5, 31, 7, 29, 13, 23, 17, 19, 7, 31, 3, 37, 11, 29, 17, 23, 5, 37, 11, 31

Offset: 4

Seiichi Manyama, Jun 13 2018

			  n  | (p,q)
  ---+----------------------------
   4 | (3,  5);
   5 | (3,  7);
   6 | (5,  7);
   7 | (3, 11);
   8 | (3, 13), (5, 11);
   9 | (5, 13), (7, 11);
  10 | (3, 17), (7, 13);
  11 | (3, 19), (5, 17);
  12 | (5, 19), (7, 17), (11, 13);

Seiichi Manyama, Rows n = 4..374, flattened
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture

Cf. A002373, A020481, A061357 (the size of row n), A078496, A078587.

row[n_] := Select[Table[{p, 2 n - p}, {p, Prime[Range[PrimePi[n]]]}], Less @@ # && AllTrue[#, PrimeQ]&] // Union;
Table[row[n], {n, 4, 25}] // Flatten (* Jean-François Alcover, Jun 16 2018 *)

A210957 Prime pair (p, q), p<=q, such that p + q = 2n and pq is the minimal product.

Original entry on oeis.org

2, 2, 3, 3, 3, 5, 3, 7, 5, 7, 3, 11, 3, 13, 5, 13, 3, 17, 3, 19, 5, 19, 3, 23, 5, 23, 7, 23, 3, 29, 3, 31, 5, 31, 7, 31, 3, 37, 5, 37, 3, 41, 3, 43, 5, 43, 3, 47, 5, 47, 7, 47, 3, 53, 5, 53, 7, 53, 3, 59, 3, 61, 5, 61, 7, 61, 3, 67, 5, 67, 3, 71, 3, 73

Offset: 2

Omar E. Pol, Jun 29 2012

nonn

A020481 and A020482 interleaved.

			-----------------------------------
                 2*n    A073046(n)
       Pair       =         =
n     (p, q)     p+q       p*q
-----------------------------------
2     (2, 2)      4          4
3     (3, 3)      6          9
4     (3, 5)      8         15
5     (3, 7)     10         21
6     (5, 7)     12         35
7     (3, 11)    14         33
8     (3, 13)    16         39
9     (5, 13)    18         65
10    (3, 17)    20         51
11    (3, 19)    22         57
12    (5, 19)    24         95

Cf. A000040, A002373, A073046, A210967.

p_n = A020481(n), n >= 2.
q_n = A020482(n), n >= 2.
p_n + q_n = 2*n, n >= 2.
p_n * q_n = A073046(n), n >= 2.

A234649 Difference between the first members of the widest and the narrowest prime pair having an arithmetic mean of n.

Original entry on oeis.org

2, 2, 4, 2, 6, 4, 6, 6, 10, 8, 12, 0, 14, 14, 10, 14, 14, 16, 18, 16, 16, 12, 22, 16, 20, 24, 24, 26, 26, 28, 26, 32, 30, 26, 36, 16, 36, 36, 28, 36, 36, 18, 44, 38, 40, 44, 42, 40, 50, 48, 40, 42, 52, 30, 42, 46, 42, 56, 56, 58, 48, 60, 64, 56, 66, 60, 48, 60, 70, 68, 68, 54, 68, 74, 60, 56

Offset: 8

Ralf Stephan, Dec 29 2013

nonn

The widest prime pair with a mean of n is (A002373(n),A020482(n)) and the narrowest is (A078587(n),A078496(n)).
Existence of a(n) for all n depends on A061357(n) > 0.
Even numbers missing in the subsequence with n<10^5 are 34,62,82,88,112,116,118,122,130,140,152...
a(n) = 0 for n=4,5,6,7,19 because A061357(n) = 1.

			The prime pairs with an arithmetic mean of 18 are (17,19), (13,23), (7,29), and (5,31), so a(18) = 17-5 = 31-19 = 12. The only pair with mean of 19 is (7,31) so a(19) = 0.

Ralf Stephan, Table of n, a(n) for n = 8..100000

Cf. A045917.

a(n)=mi=0;ma=0;forprime(p=3,n-1,if(isprime(2*n-p),if(!mi,mi=2*n-p);ma=2*n-p));if(!ma,-1,mi-ma)

a(n) = A078587(n) - A002373(n) = A078496(n) - A020482(n).

A235859 Define a(4)=3, then a(n+1) is the smallest prime P such that a(n) <= P < 2n with 2n-P=Q prime and, if not possible, a(n+1) is the smallest prime P such that P < a(n) < 2n with 2n-P=Q prime.

Original entry on oeis.org

3, 3, 5, 11, 11, 11, 13, 17, 17, 19, 23, 23, 29, 29, 29, 31, 37, 37, 37, 41, 41, 43, 47, 47, 53, 53, 53, 59, 59, 59, 61, 67, 67, 67, 71, 71, 73, 79, 79, 79, 83, 83, 89, 89, 89, 19, 29, 29, 31, 47, 47, 67, 71, 71, 73, 89, 89, 103, 107, 107, 109, 113, 113

Offset: 4

Pierre CAMI, Jan 16 2014

			a(4)=3 as 2*4-3=5 prime by definition
a(5)=3 as 2*5-3=7 prime, a(5)=a(4), a(5)<5
a(6)=5 as 2*6-5=7 prime, a(6)>a(5), a(6)<6
a(7)=5 not possible as 14-5=9 composite
a(7)=7 not possible as 7=7
a(7)=11 as 2*7-11=3 prime
.........................
a(48)=89 as 2*48-89=7 prime
a(49)=89 not possible as 2*49-89=9 composite
a(49)=97 not possible as 2*49-97=unity
a(49)=19 as 19 is the smallest prime such that 2*49-19 is prime
a(50)=29 as 29 is the smallest prime >=19 such that 2*50-29 is prime

Pierre CAMI, Table of n, a(n) for n = 4..10005

Cf. A002373, A020483, A235649.

A245077 Largest k such that the smallest prime satisfying Goldbach's conjecture is less than or equal to (2n)^(1/k).

Original entry on oeis.org

2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 3, 3, 2, 1, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 2, 4, 2, 4, 2, 2, 4, 2, 2, 1, 4, 2, 4, 4, 2, 4, 4, 2, 4, 2, 2, 1, 2, 1, 1, 4

Offset: 2

Jon Perry, Jul 11 2014

nonn

The 1's appear as in A244408.

			For n=5 we have 3+7=10. As rt3(10)<3<sqrt(10), a(5)=2.

Jens Kruse Andersen, Table of n, a(n) for n = 2..10000

Cf. A244408.

Cf. A020481, A002373, A025018, A025019.

for (n=2, 100, p=2; while(!isprime(2*n-p), p=nextprime(p+1)); k=1; while(p<=(2*n)^(1/k), k++); print1(k-1", ")) \\ Jens Kruse Andersen, Jul 12 2014

Definition corrected by Jens Kruse Andersen, Jul 12 2014

A274189 Even numbers 2n that satisfy an extended Goldbach conjecture: They have at least one Goldbach partition 2n = p + q (p <= q; p, q prime) that satisfies p <= sqrt(n), at least one with sqrt(n) < p <= sqrt(2n) and at least one with p > sqrt(2n).

Original entry on oeis.org

34, 46, 50, 66, 74, 78, 86, 138, 142, 160, 162, 168, 170, 174, 176, 178, 180, 184, 186, 194, 202, 204, 206, 234, 236, 238, 240, 242, 246, 252, 254, 264, 270, 276, 282, 284, 290, 294, 296, 298, 300, 310, 320, 324, 328, 334, 348, 354, 364, 366, 370, 372, 376, 378, 384, 386, 390, 392, 396, 400

Offset: 1

Corinna Regina Böger, Dec 11 2016

nonn

This sequence contains all even numbers that are not in A279040 or in A273457. I have verified numerically for all even numbers 4 < 2n < 4*10^10 that a Goldbach partition with the additional condition p > sqrt(2n) exists. It is conjectured that a(n) = 2*(n+12987) for all n > 7838315. If this conjecture is true, all even numbers 2n > 15702604 have all three types of Goldbach partitions and therefore satisfy the "extended Goldbach conjecture".

			a(1) = 34 = 3 + 31 = 5 + 29 = 11 + 23 = 17 + 17. Since 3 < sqrt(17) < 5 < sqrt(34) < 11 < 17, all three types of Goldbach partitions exist for 34.

Corinna Regina Böger, Table of n, a(n) for n = 1..100000
Index entries for sequences related to Goldbach conjecture

Cf. A002373, A020481, A025018, A025019, A244408, A273457, A279040.

GoldbachRange(n,mn,mx)=forprime(p=mn,mx, if(isprime(n-p), return(1))); 0
is(n)=n%2==0 && GoldbachRange(n, 2, sqrtint(n/2)) && GoldbachRange(n, sqrtint(n/2-1)+1, sqrtint(n)) && GoldbachRange(n, sqrtint(n-1)+1, n/2) \\ Charles R Greathouse IV, Dec 16 2016

A335045 Minimal common prime of two Goldbach partitions of 2n and 2(n+1) or zero if no common prime exists.

Original entry on oeis.org

0, 3, 3, 5, 7, 3, 5, 7, 3, 5, 7, 23, 11, 13, 3, 5, 7, 0, 11, 13, 3, 5, 7, 47, 11, 13, 53, 17, 19, 3, 5, 7, 0, 11, 13, 3, 5, 7, 0, 11, 13, 83, 17, 19, 89, 23, 37, 0, 29, 31, 3, 5, 7, 3, 5, 7, 113, 11, 13, 0, 17, 19, 0, 23, 31, 131, 29, 31, 3, 5, 7, 0, 11, 13, 3, 5, 7, 0, 11, 13, 0, 17, 19, 167, 23, 37, 173

Offset: 2

Ivan N. Ianakiev, May 21 2020

nonn

a(n) is the least prime p such that 2n-p is in A001359, or 0 if no such p exists. - Robert Israel, May 21 2020

			4 = 2+2 and 6 = 3+3. Since those are the only available Goldbach partitions and they have no common prime, a(4/2) = a(2) = 0.
14 = 3+11 and 16 = 3+13, so a(14/2) = a(7) = 3.

Index entries for sequences related to Goldbach conjecture

Cf. A001359, A002372, A002373, A002375, A045917, A335046.

N:= 100:
P:= select(isprime, {seq(i,i=3..2*N-1,2)}):
T:= P intersect map(`-`,P,2):
f:= n -> subs(infinity=0, min(P intersect map(t -> 2*n-t, T))):
map(f, [$2..N]); # Robert Israel, May 21 2020

d[n_]:=Flatten[Cases[FrobeniusSolve[{1,1},2*n],{?PrimeQ}]]
e[n_]:=Intersection[d[n],d[n+1]]; f[n_]:=If[e[n]=={},0,Min[e[n]]];f/@Range[2,100]

A335046 Maximal common prime of two Goldbach partitions of 2n and 2(n+1) or zero (if common prime does not exist).

Original entry on oeis.org

0, 3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 19, 29, 31, 31, 0, 37, 37, 41, 43, 43, 47, 47, 43, 53, 53, 43, 59, 61, 61, 0, 67, 67, 71, 73, 73, 0, 79, 79, 83, 83, 79, 89, 89, 79, 0, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 109, 0, 113, 109, 0, 127, 127, 131, 131, 127, 137, 139, 139

Offset: 2

Ivan N. Ianakiev, May 21 2020

nonn

			4 = 2+2 and 6 = 3+3. Since those are the only available Goldbach partitions and they have no common prime, a(4/2) = a(2) = 0. 14 = 3+11 and 16 = 5+11, so a(14/2) = a(7) = 11.

Index entries for sequences related to Goldbach conjecture

Cf. A002372, A002373, A002375, A045917, A060308, A335045.

S:= proc(n) option remember; {seq((h-> `if`(
      andmap(isprime, h), h, [])[])([n+i, n-i]), i=0..n-2)}
    end:
a:= n-> max(0, (S(n) intersect S(n+1))[]):
seq(a(n), n=2..80);  # Alois P. Heinz, Jun 20 2020

d[n_]:=Flatten[Cases[FrobeniusSolve[{1,1},2*n],{?PrimeQ}]]
e[n_]:=Intersection[d[n],d[n+1]]; f[n_]:=If[e[n]=={},0,Max[e[n]]];
f/@Range[2,100]

A083372 Least number having exactly two odd prime factors that differ by 2n.

Original entry on oeis.org

15, 21, 55, 33, 39, 85, 51, 57, 115, 69, 203, 145, 87, 93, 259, 185, 111, 205, 123, 129, 235, 141, 371, 265, 159, 413, 295, 177, 183, 469, 335, 201, 355, 213, 219, 553, 395, 237, 415, 249, 623, 445, 267, 1313, 679, 485, 291, 505, 303, 309, 535, 321, 327, 565

Offset: 1

Lekraj Beedassy, Jun 05 2003

nonn

The lesser of the two factors is in A002373.

			We have a(4) = 33 because 33 = 3*11, with 11 - 3 = 2*4, the smallest number with this property. Others are 85 = 5*13, 209 = 11*19, 713 = 23*31, 1073 = 29*37, 3233 = 53*61, ...

Cf. A046388.

f[n_] := Block[{p = 3}, While[ ! PrimeQ[p] || ! PrimeQ[p + 2n], p++ ]; p(p + 2n)]; Table[ f[n], {n, 1, 55}]
Table[#(#+2n)&/@Select[Prime[Range[2,100]],PrimeQ[#+2n]&,1],{n,60}]// Flatten (* Harvey P. Dale, May 26 2018 *)

Edited and extended by Robert G. Wilson v, Jun 07 2003

cp's OEIS Frontend

A273457 Even numbers 2n that do not have a Goldbach partition 2n = p + q (p < q; p, q prime) satisfying sqrt(n) < p <= sqrt(2n).

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A305883 Triangle read by rows: row n lists the pairs (p, q) such that p, q are primes, p+q=2*n and p < q.

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A210957 Prime pair (p, q), p<=q, such that p + q = 2*n and p*q is the minimal product.

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A234649 Difference between the first members of the widest and the narrowest prime pair having an arithmetic mean of n.

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A235859 Define a(4)=3, then a(n+1) is the smallest prime P such that a(n) <= P < 2*n with 2*n-P=Q prime and, if not possible, a(n+1) is the smallest prime P such that P < a(n) < 2*n with 2*n-P=Q prime.

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A245077 Largest k such that the smallest prime satisfying Goldbach's conjecture is less than or equal to (2n)^(1/k).

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Extensions

A274189 Even numbers 2n that satisfy an extended Goldbach conjecture: They have at least one Goldbach partition 2n = p + q (p <= q; p, q prime) that satisfies p <= sqrt(n), at least one with sqrt(n) < p <= sqrt(2n) and at least one with p > sqrt(2n).

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A335045 Minimal common prime of two Goldbach partitions of 2n and 2(n+1) or zero if no common prime exists.

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Mathematica

A335046 Maximal common prime of two Goldbach partitions of 2n and 2(n+1) or zero (if common prime does not exist).

A210957 Prime pair (p, q), p<=q, such that p + q = 2n and pq is the minimal product.

A235859 Define a(4)=3, then a(n+1) is the smallest prime P such that a(n) <= P < 2n with 2n-P=Q prime and, if not possible, a(n+1) is the smallest prime P such that P < a(n) < 2n with 2n-P=Q prime.