A025018 -id:A025018 - cp's OEIS frontend

A025019 Smallest prime in Goldbach partition of A025018(n).

2, 3, 5, 7, 19, 23, 31, 47, 73, 103, 139, 173, 211, 233, 293, 313, 331, 359, 383, 389, 523, 601, 727, 751, 829, 929, 997, 1039, 1093, 1163, 1321, 1427, 1583, 1789, 1861, 1877, 1879, 2029, 2089, 2803, 3061, 3163, 3457, 3463, 3529, 3613, 3769, 3917, 4003, 4027, 4057

Offset: 1

David W. Wilson, Dec 11 1999

nonn

Increasing subsequence of A020481.

For n > 2, a(n) ~ (log(A025018(n)))^e/e, while an upper bound could be written as UB(a(n)) = floor(log(A025018(n)))^e/2 (therefore, for any even v such that 12 <= v <= A025018(67) UB is true). It looks that both approximation and UB are true for any n > 2. Assuming the second equation to be true, UB(10^80) = 718967, UB(10^500) = 104745517, etc. - Sergey Pavlov, Jan 17 2021

			1427 and 1583 are two consecutive terms because A020481(167535419) = 1427 and A020481(209955962) = 1583 and for 167535419 < n < 209955962 A020481(n) <= 1427.

N. J. A. Sloane, Table of n, a(n) for n=1..67 (from the web page of Tomás Oliveira e Silva)
Mark A. Herkommer, Goldbach Conjecture Research
Tomás Oliveira e Silva, Goldbach conjecture verification
Jörg Richstein, Verifying the Goldbach conjecture up to 4 * 10^14, Math. Comp., 70 (2001), 1745-1749.
Index entries for sequences related to Goldbach conjecture

Cf. A025018, A020481, A097224, A097226.

p = 1; q = {}; Do[ k = 2; While[ !PrimeQ[k] || !PrimeQ[2n - k], k++ ]; If[k > p, p = k; q = Append[q, p]], {n, 2, 10^8}]; q

Edited and extended by Robert G. Wilson v, Dec 13 2002

More terms and b-file added by N. J. A. Sloane, Nov 28 2007

A244408 Even numbers 2k such that the smallest prime p satisfying p+q=2k (q prime) is greater than or equal to sqrt(2k).

Original entry on oeis.org

4, 6, 8, 12, 18, 24, 30, 38, 98, 122, 126, 128, 220, 302, 308, 332, 346, 488, 556, 854, 908, 962, 992, 1144, 1150, 1274, 1354, 1360, 1362, 1382, 1408, 1424, 1532, 1768, 1856, 1928, 2078, 2188, 2200, 2438, 2512, 2530, 2618, 2642, 3458, 3818, 3848

Offset: 1

Jon Perry, Jun 27 2014

nonn

a(74) = 63274 is probably the last term. Oliveira e Silva's work shows there are no more terms below 4*10^18. The largest p below that is p = 9781 for 2k = 3325581707333960528, where sqrt(2k) = 1823617752. - Jens Kruse Andersen, Jul 03 2014

The sequence definition is equivalent to: "Even integers k such that there exists a prime p with p = min{q: q prime and (k-q) prime} and k <= p^2" and therefore this is a member of the EGN-family (Cf. A307782). - Corinna Regina Böger, May 01 2019

			The smallest prime for 38 is 7, and 7 >= sqrt(38).

Jens Kruse Andersen, Table of n, a(n) for n = 1..74
Tomás Oliveira e Silva, Goldbach conjecture verification
Index entries for sequences related to Goldbach conjecture

Cf. A020481, A002373, A025018, A025019, A306746, A307782.

a244408 n = a244408_list !! (n-1)
a244408_list = map (* 2) $ filter f [2..] where
   f x = sqrt (fromIntegral $ 2 * x) <= fromIntegral (a020481 x)
-- Reinhard Zumkeller, Jul 07 2014

for(n=1, 50000, forprime(p=2, n, if(isprime(2*n-p), if(p>=sqrt(2*n), print1(2*n", ")); break))) \\ Jens Kruse Andersen, Jul 03 2014

A025017 a(n) = least 2k such that p is the least prime in a Goldbach partition of 2k, where p = prime(n).

Original entry on oeis.org

4, 6, 12, 30, 124, 122, 418, 98, 220, 346, 308, 1274, 1144, 962, 556, 2512, 3526, 1382, 1856, 4618, 992, 3818, 7432, 12778, 5978, 26098, 2642, 23266, 10268, 19696, 6008, 34192, 22606, 5372, 37768, 13562, 9596, 22832, 59914, 7426, 88786, 50312, 97768

Offset: 1

David W. Wilson

nonn

Minimal integer m such that m=p(n)+q=sum of 2 primes, where p(n)<=q is the n-th prime and there is no prime rRobin Garcia, Feb 12 2005

The increasing subsequence k(n), such that for all m>n, k(m)>k(n) is A025018, and the associated sequence of primes is A025019. - David James Sycamore, Feb 05 2018

			a(4)=30=7+23 because p(4)=7, q=23 is prime and there is no prime r<p(4)=7 such that a(4)-r is prime.

N. J. A. Sloane, Table of n, a(n) for n = 1..977 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, Goldbach conjecture verification
Index entries for sequences related to Goldbach conjecture

For records see A133427, A133428.

Cf. A025018, A025019.

p1 = primes(1000000); d(1, :) = p1; d(2, :) = d(1, :) - d(1, :); i = 4; k = 1; n = 0; while i <= 5000000 while not(isprime(i - d(1, k))) k = k + 1; end; if d(2, k) == 0 d(2, k) = i; if k == n + 1 while d(2, n+1) > 0 n = n + 1; end; if n > 0 d(2, 1:n) end; end; end; k = 1; i = i + 2; end; - Lei Zhou, Jan 26 2005

Gold(n)=forprime(p=2,n,if(isprime(n-p),return(p)))
a(n,p=prime(n))=my(k=2); while(Gold(k+=2)!=p,); k \\ Charles R Greathouse IV, Sep 28 2015

Edited by N. J. A. Sloane, May 05 2007; b-file added Nov 27 2007

A152522 a(n) is the least even number such that if p_i is the i-th prime then a(n)-p_i, i=1..n, are composite numbers.

Original entry on oeis.org

6, 12, 30, 98, 98, 98, 98, 220, 308, 308, 556, 556, 556, 556, 992, 992, 992, 992, 992, 992, 2642, 2642, 2642, 2642, 2642, 2642, 5372, 5372, 5372, 5372, 5372, 5372, 5372, 7426, 7426, 7426, 7426, 7426, 7426, 43532, 43532, 43532, 43532, 43532, 43532

Offset: 1

Vladimir Shevelev, Dec 06 2008

nonn

Conjecture (A. Granville, H. te Riele and J. van de Lune, 1989) Let, for even N, p=p(N) be the least prime such that N-p is prime as well. Then p(N)=O((log(N))^2log(log(N))). [Vladimir Shevelev, Dec 08 2008]

It is conjectured that a(n) >= prime(k+1) + 3. See Conjecture C p. 2 in Phong & Dongdong. - Michel Marcus, Aug 02 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Andrew Granville, J. Van de Lune, and Herman te Riele, Checking the Goldbach Conjecture on a vector computer, Number Theory and Applications (1989), pp. 423-434.
Bui Minh Phong, Li Dongdong, Elementary problems which are equivalent to the Goldbach’s conjecture, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 33-37.

Cf. A025018.

Bag={};
f[n_]:=NestList[NextPrime,2,n];
GoldbachTest[n_?EvenQ,p_List]:=Block[{m=Length[p],i=1},While[i<=m&&CompositeQ[n-p[[i]]],i+=1];If[i>m,{0,0},{#,n-#}&[p[[i]]]]];
Do[n=4;While[(GoldbachTest[n,f[j]])!={0,0},{n=n+2}];AppendTo[Bag,n],{j,0,44}]; Bag  (* Gilmar Rodriguez Pierluissi, Aug 23 2018 *)

a(n)=my(P=prime(n));forstep(k=6,9e99,2,forprime(p=3,P,if(isprime(k-p),next(2)));return(k)) \\ Charles R Greathouse IV, Sep 04 2011

A279040 Even numbers 2k such that the smallest prime p satisfying p+q=2k (q prime) is greater than or equal to sqrt(k).

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 24, 28, 30, 36, 38, 42, 48, 54, 60, 68, 80, 90, 96, 98, 122, 124, 126, 128, 148, 150, 190, 192, 208, 210, 212, 220, 222, 224, 302, 306, 308, 326, 330, 332, 346, 368, 398, 418, 458, 488, 518, 538, 540, 542, 556, 640, 692, 710, 796, 854, 908, 962, 968, 992, 1006

Offset: 1

Corinna Regina Böger, Dec 04 2016

nonn

a(n) is an extension of A244408.
It is conjectured that a(230) = 503222 is the last term. Oliveira e Silva's work shows that there are no more terms below 4*10^18.
The sequence definition is equivalent to: "Even integers k such that there exists a prime p with p = min{q: q prime and (k - q) prime} and k < 2*p^2" and therefore this is a member of the EGN- family (Cf. A307782). - Corinna Regina Böger, May 01 2019

			The smallest prime for 42 is 5 with 5 > sqrt(21), but not smaller than sqrt(42), and therefore 42 does not belong to A244408. The smallest prime for 38 is 7, and 7 >= sqrt(38), and therefore 38 also belongs to A244408.

Corinna Regina Böger, Table of n, a(n) for n = 1..230
Tomás Oliveira e Silva, Goldbach conjecture verification
Index entries for sequences related to Goldbach conjecture

Cf. A002373, A020481, A025018, A025019, A093161, A244408, A307542, A307782

Select[Range[4, 1006, 2], Function[n, Select[#, PrimeQ@ Last@ # &][[1, 1]] >= Sqrt[n/2] &@ Map[{#, n - #} &, Prime@ Range@ PrimePi@ n]]] (* Michael De Vlieger, Dec 06 2016 *)

isok(n) = forprime(p=2, n, if (isprime(n-p), if (p >= sqrt(n/2), return(1), return(0))));
lista(nn) = forstep(n=2, nn, 2, if (isok(n), print1(n, ", "))) \\ Michel Marcus, Dec 04 2016

A093161 Even integers k such that there exists a prime p with p = min{q: q prime and (k - q) prime} and (k - p) < p^3.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 36, 38, 42, 48, 52, 54, 58, 60, 66, 68, 72, 78, 80, 84, 88, 90, 94, 96, 98, 102, 108, 114, 118, 120, 122, 124, 126, 128, 138, 146, 148, 150, 158, 164, 174, 180, 188, 190, 192, 206, 208, 210, 212, 218, 220, 222, 224

Offset: 1

Jason Earls, May 10 2004

It is conjectured that the sequence is finite with last term a(104820) = 5714500178 and it is proven that there are no more terms below 4*10^18. This is an extension of A307542. - Corinna Regina Böger, Apr 14 2019

			63274 is in the sequence because 63274 = 293 + 62981 is the Goldbach partition with the smallest prime and 293^3 = 25153757 is > 62981. [clarified by _Corinna Regina Böger_, Apr 22 2019]

Corinna Regina Böger, Table of n, a(n) for n = 1..10000
Corinna Regina Böger, a-file, Table of n, a(n) for n=1..104820
John F. Nash, Jr., Goldbach Programs

Cf. A025018.

isS := proc(n) local p; for p from 2 while p^3 < (n-p) do
if isprime(p) and isprime(n-p) then return false fi od; true end:
isa := n -> irem(n,2) = 0 and isS(n): select(isa, [$4..224]); # Peter Luschny, Apr 26 2019

okQ[n_] := Module[{p}, For[p = 2, p <= n/2, p = NextPrime[p], If[p^3 + p < n && PrimeQ[n - p], Return[False]]]; True];
Select[Range[4, 250, 2], okQ] (* Jean-François Alcover, Jun 11 2019, from PARI *)

noSpecialGoldbach(n) = forprime(p=2, n/2, if(p^3+p2 && n%2 == 0 && noSpecialGoldbach(n) \\ Corinna Regina Böger, Apr 14 2019

New name by Corinna Regina Böger, Apr 27 2019

A129302 Where records occur in A082467.

Original entry on oeis.org

4, 5, 7, 11, 19, 43, 97, 146, 163, 191, 223, 344, 457, 526, 877, 904, 1049, 1114, 1307, 1736, 1751, 1781, 2129, 2476, 3097, 3551, 5131, 8504, 10342, 10357, 18233, 24776, 40072, 68707, 99719, 125903, 174913, 181267, 371428, 827576, 936118, 1054141

Offset: 1

Klaus Brockhaus, Apr 08 2007

nonn

			The sixth record in A082467 is A129301(6) = 24 = A082467(43), hence a(6) = 43.

T. D. Noe, Table of n, a(n) for n=1..69

Cf. A082467, A129301 (records).

Cf. A025018

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)

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

cp's OEIS Frontend

A025019 Smallest prime in Goldbach partition of A025018(n).

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A244408 Even numbers 2k such that the smallest prime p satisfying p+q=2k (q prime) is greater than or equal to sqrt(2k).

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A025017 a(n) = least 2k such that p is the least prime in a Goldbach partition of 2k, where p = prime(n).

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A152522 a(n) is the least even number such that if p_i is the i-th prime then a(n)-p_i, i=1..n, are composite numbers.

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A279040 Even numbers 2k such that the smallest prime p satisfying p+q=2k (q prime) is greater than or equal to sqrt(k).

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A093161 Even integers k such that there exists a prime p with p = min{q: q prime and (k - q) prime} and (k - p) < p^3.

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Maple

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A129302 Where records occur in A082467.

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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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