A020481 -id:A020481 - cp's OEIS frontend

A236569 Least term p of A236568 with 2*n - p prime, or 0 if such a prime p does not exist.

0, 0, 3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 11, 3, 3, 5, 31, 3, 5, 3, 3, 5, 3, 5, 11, 3, 5, 31, 3, 3, 5, 31, 3, 5, 3, 3, 5, 43, 3, 5, 3, 5, 11, 3, 5, 43, 31, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 11, 43, 11, 43, 31, 3, 5, 3, 5, 11, 3

Offset: 1

Zhi-Wei Sun, Jan 29 2014

nonn

The conjecture in A236566 implies that a(n) > 0 for all n > 2.

			a(3) = 3 since prime(3 + 2) + 2 = 11 + 2 = 13 and 2*3 - 3 = 3 are both prime, but prime(2 + 2) + 2 = 9 is not.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A020481, A236566, A236568.

p[m_]:=PrimeQ[Prime[m+2]+2]
Do[Do[If[p[Prime[k]]&&PrimeQ[2n-Prime[k]],Print[n," ",Prime[k]];Goto[aa]],{k,1,PrimePi[2n-1]}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]

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).

Original entry on oeis.org

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)

A284928 Numbers k such that 2k + p is composite for all primes p, q with p + q = 2k.

Original entry on oeis.org

0, 1, 2, 3, 14, 19, 26, 29, 31, 34, 37, 40, 41, 44, 47, 49, 56, 59, 61, 62, 64, 68, 73, 74, 76, 79, 82, 83, 86, 89, 91, 92, 94, 95, 103, 104, 106, 107, 109, 110, 112, 119, 121, 122, 124, 125, 128, 131, 134, 139, 142, 145, 146, 148, 149, 151, 152, 154, 158, 160, 161, 163, 164, 166, 169

Offset: 1

M. F. Hasler, Apr 06 2017

nonn

Or, numbers k such that there is no prime p < 2k with 2k - p and 2k + p both prime.
The two initial terms vacuously satisfy the definition, but all even numbers >= 4 are the sum of two primes, according to the Goldbach conjecture.
See also A284919, twice this sequence, which lists the values of 2k.

Claudio Meller and others, New sequence, SeqFan list, April 5, 2017. (Click "next" for subsequent contributions.)

Cf. A284919 (twice this), A002375 (number of decompositions p + q = 2k), A020481 (least p: p + q = 2k).

is_A284928(n)=!forprime(p=2,n, isprime(2*n-p) && (isprime(2*n+p) || isprime(4*n-p)) && return) \\ M. F. Hasler, Apr 06 2017

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 *)

A112826 Conjectured values of A112825 which are 0.

Original entry on oeis.org

58, 62, 82, 88, 108, 112, 114, 116, 118, 122, 130, 140, 148, 152, 162, 182, 184, 196, 198, 200, 202, 212, 214, 218, 240, 242, 244, 250, 254, 256, 258, 262, 272, 282, 284, 292, 296, 298, 316, 320, 322, 332, 336, 340, 358, 362, 366, 382, 394, 400, 410, 412

Offset: 1

Robert G. Wilson v, Sep 05 2005

nonn

It is conjectured that there does not exist a Goldbach partition yielding a Goldbach "gap" of n as defined, for n=58,62,82,....

These are the even numbers that do not appear in A112824.

Cf. A020481.

f[n_] := Block[{p = 2, q = n/2}, While[ !PrimeQ[p] || !PrimeQ[n - p], p++ ]; While[ !PrimeQ[q] || !PrimeQ[n - q], q-- ]; q - p];
t = Table[0, {10000}];
Do[a = f[2n]; If[a < 10000 && t[[a/2 + 1]] == 0, t[[a/2 + 1]] = 2n], {n, 2, 10^6}];
Take[ 2*Flatten[ Position[t, 0] - 1], 52]

Corrected by T. D. Noe, Feb 14 2011

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.

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

A255274 From Goldbach conjecture: Consider the pairs (2n-+1, 3), (2n-1, 5), (2n-3, 7), ..., (3, 2n+1) of odd numbers having sum 2n+4; a(n) is the index of the first pair of primes (p, q) on the list.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 9, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 6, 5, 6, 9, 1, 2, 1, 2, 3, 1, 1, 2, 3, 5, 5, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2

Offset: 1

Michel Lagneau, Feb 20 2015

nonn

a(n) = A049847(n) for n = 1..46. The values of n such that a(n) is different from A049847(n) are 47, 59, 62, 72, 93, 102, 108, 123, 144, 149, 152, 161, 164, 171, 182, 197, 203, 207, 213, 227, ...

The corresponding pairs of primes are (3, 3), (3, 5), (3, 7), (5, 7), (3, 11), (3, 13), (5, 13), (3, 17), ... (A210957).

			a(13)=3 because 2*13 + 4 = 30 => 13 pairs (27,3), (25,5), (23,7), ..., (3,27) and the pair (23,7) is the third pair having prime elements.

Michel Lagneau, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A255274, Sequence Machine.

Cf. A000040, A002375, A049847, A210957, A020482.

nn:=100:for n from 6 by 2 to nn do:ii:=0:it:=1:for p from 3 by 2 to n while(ii=0) do:if type(n-p,prime)=true and type(p,prime)=true then ii:=1: printf(`%d, `,it):else it:=it+1:fi:od:od:

a(n)=my(m=2*n+4); forprime(q=3, n+2, if(isprime(m-q), return(q\2))) \\ Charles R Greathouse IV, Jan 07 2022

a(n) = n + (3-A020482(n+2))/2 = (A020481(n+2)-1)/2 via the Maiga link. - Bill McEachen, Jan 02 2022

Edited by N. J. A. Sloane, Sep 12 2017

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

A112827 Least value k which is the beginning of a null Goldbach chain of length exactly n.

Original entry on oeis.org

60, 184, 242, 114, 2314, 1382

Offset: 1

Robert G. Wilson v, Sep 05 2005

The first even number of A112826(k/2) consisting of a run of n zeros long.

			a(1)=60; a(2)=184 because A112825(92) and A112825(93)=0 but A112825(91) and A112825(94) are not equal to 0.
a(3)=242 because A112825(121), A112825(122) and A112825(123)=0 but A112825(120) and A112825(124) are not equal to 0.

Cf. A020481.

f[n_] := Block[{p = 2, q = n/2}, While[ !PrimeQ[p] || !PrimeQ[n - p], p++ ]; While[ !PrimeQ[q] || !PrimeQ[n - q], q-- ]; q - p]; t = Table[0, {10000}]; Do[a = f[2n]; If[a < 10000 && t[[a + 1]] == 0, t[[a + 1]] = 2n], {n, 2, 10^6}]; g = Flatten[ Position[t, 0]];

cp's OEIS Frontend

A236569 Least term p of A236568 with 2*n - p prime, or 0 if such a prime p does not exist.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A284928 Numbers k such that 2k + p is composite for all primes p, q with p + q = 2k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A112826 Conjectured values of A112825 which are 0.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A255274 From Goldbach conjecture: Consider the pairs (2n-+1, 3), (2n-1, 5), (2n-3, 7), ..., (3, 2n+1) of odd numbers having sum 2n+4; a(n) is the index of the first pair of primes (p, q) on the list.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

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