A025019 -id:A025019 - cp's OEIS frontend

A002373 Smallest prime in decomposition of 2n into sum of two odd primes.

3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 3, 3, 5, 7, 3, 5, 3, 3, 5, 3, 5, 7, 3, 5, 7, 3, 3, 5, 7, 3, 5, 3, 3, 5, 7, 3, 5, 3, 5, 7, 3, 5, 7, 19, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 13, 11, 13, 19, 3, 5, 3, 5, 7, 3, 3, 5, 7, 11, 11, 3, 3, 5, 7, 3, 5, 7, 3, 5, 3, 5, 7, 3, 5, 7, 3, 3, 5, 7, 11, 11, 3, 3, 5, 3

Offset: 3

N. J. A. Sloane

See A020481 for another version.
a(A208662(n)) = A065091(n) and a(m) <> A065091(n) for m < A208662(n). - Reinhard Zumkeller, Feb 29 2012
Records are in A025019, their indices in A051610. - Ralf Stephan, Dec 29 2013
Note that these primes do not all belong to a twin prime pair. The first instance is a(110) = 23. - Michel Marcus, Aug 17 2020 from a suggestion by Pierre CAMI

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 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).

T. D. Noe, Table of n, a(n) for n = 3..10000
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A002372, A002374, A014092, A065091, A010051.

a002373 n = head $ dropWhile ((== 0) . a010051 . (2*n -)) a065091_list -- Reinhard Zumkeller, Feb 29 2012

Table[k = 2; While[q = Prime[k]; ! PrimeQ[2*n - q], k++]; q, {n, 3, 100}] (* Jean-François Alcover, Apr 26 2011 *)
Table[Min[Flatten[Select[IntegerPartitions[2*n,{2}],AllTrue[ #,OddQ] && AllTrue[#,PrimeQ]&]]],{n,3,100}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 31 2020 *)

a(n)=forprime(p=3,n,if(isprime(2*n-p), return(p))) \\ Charles R Greathouse IV, May 18 2015

More terms from Ray Chandler, Sep 19 2003

A025018 Numbers k such that least prime in the Goldbach partition of k increases.

Original entry on oeis.org

4, 6, 12, 30, 98, 220, 308, 556, 992, 2642, 5372, 7426, 43532, 54244, 63274, 113672, 128168, 194428, 194470, 413572, 503222, 1077422, 3526958, 3807404, 10759922, 24106882, 27789878, 37998938, 60119912, 113632822, 187852862, 335070838

Offset: 1

David W. Wilson

Charles R Greathouse IV, Table of n, a(n) for n = 1..69 (from Tomás Oliveira e Silva)
Mark A. Herkommer, Goldbach Conjecture Research
Jorg Richstein, Verifying Goldbach's Conjecture up to 4 * 10^14, Math. of Computation, Vol. 70, No. 236, pp. 1745-1749 (July 2000)
Index entries for sequences related to Goldbach conjecture
Tomás Oliveira e Silva, Goldbach conjecture verification

Cf. A001172, A025019.

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

Gold(n)=forprime(p=2,min(n\2,default(primelimit)),if(isprime(n-p),return(p)))
r=0;forstep(n=4,1e6,2,t=Gold(n);if(t>r,r=t;print1(n", "))) \\ Charles R Greathouse IV, Feb 21 2012

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

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

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

A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.

Original entry on oeis.org

6, 8, 8, 10, 12, 14, 18, 24, 32, 48, 72, 110, 174, 274, 438, 704, 1134, 1830, 2952, 4762, 7698, 12450, 20128, 32560, 52660, 85168, 137752, 222844, 360564, 583392, 943902, 1527222, 2471074, 3998274, 6469334, 10467566, 16936850, 27404300, 44341050, 71745324

Offset: 1

Vladimir Shevelev, Mar 16 2013

nonn

Conjecture. lim a(n+1)/a(n)=phi as n goes to infinity (phi=golden ratio).

			Let n=6. Since a(4) = 10, a(5) = 12 and g(10) = g(12) = 7, then a(6) = 7 + 7 = 14.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000045, A002375, A025019, A216835.

a[1] = 6; a[2] = 8; g[n_] := Module[{tmp,k=1}, While[!PrimeQ[n-(tmp=NextPrime[n,-k])], k++]; tmp]; a[n_] := a[n] = g[a[n-1]] + g[a[n-2]]; Table[a[n], {n,1,100}]

For n>=5, a(n) = A216835(n-3) + A216835(n-4).

A216835 Fibonacci + Goldbach (dual sequence to A216275). a(1)=5, a(2)=7 and for n>=3, a(n) = g(a(n-1) + a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.

Original entry on oeis.org

5, 7, 7, 11, 13, 19, 29, 43, 67, 107, 167, 271, 433, 701, 1129, 1823, 2939, 4759, 7691, 12437, 20123, 32537, 52631, 85121, 137723, 222841, 360551, 583351, 943871, 1527203, 2471071, 3998263, 6469303, 10467547, 16936753, 27404297, 44341027, 71745313, 116086303

Offset: 1

Vladimir Shevelev, Mar 16 2013

nonn

Conjecture. lim a(n+1)/a(n)=phi as n goes to infinity (phi=golden ratio).

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000045, A002375, A025019, A216275.

a[1] = 5; a[2] = 7; g[n_] := Module[{tmp,k=1}, While[!PrimeQ[n-(tmp=NextPrime[n,-k])], k++]; tmp]; a[n_] := a[n] = g[a[n-1] + a[n-2]]; Table[a[n], {n,1,100}]

a(n) = g(A216275(n+2)).

A129301 Records in A082467.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 33, 36, 42, 60, 75, 84, 87, 90, 93, 102, 117, 120, 135, 138, 168, 180, 183, 210, 228, 300, 333, 369, 474, 486, 525, 621, 720, 792, 810, 846, 1086, 1281, 1305, 1515, 1590, 1617, 1722, 1794

Offset: 1

Klaus Brockhaus, Apr 08 2007

nonn

			As can be gathered from A082467, the first six records are A082467(4) = 1, A082467(5) = 2, A082467(7) = 4, A082467(11) = 6, A082467(19) = 12, A082467(43) = 24. Hence a(1) to a(6) are 1, 2, 4, 6, 12, 24.

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

Cf. A082467, A129302 (where records occur).

Cf. A025019

a(n) = A082467(A129302(n)). - Jason Kimberley, Sep 04 2011

A097224 Nondecreasing subsequence of A020481.

Original entry on oeis.org

2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 19, 19, 23, 31, 31, 47, 73, 103, 139, 173, 173, 173, 211, 233, 293, 313, 331, 359, 383, 389, 389, 523, 601, 727, 751, 829, 929, 997, 1039, 1093, 1163, 1321, 1427, 1583

Offset: 1

Farideh Firoozbakht, Aug 09 2004

nonn

Cf. A020481, A051610, Union(A097224)=A025019.

c[n_] := Block[{m = 2}, While[ !PrimeQ[2n - Prime[m]], m++ ]; Prime[m]]; v={2}; Do[ p = c[n]; If[ p >= v1, v1 = p; AppendTo[v, p]; Print[p]], {n, 3, 215000000}]; v

More terms from Robert G. Wilson v, Aug 10 2004

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)

cp's OEIS Frontend

A002373 Smallest prime in decomposition of 2n into sum of two odd primes.

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A025018 Numbers k such that least prime in the Goldbach partition of k increases.

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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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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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A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2*m) is the maximal prime p < 2*m such that 2*m - p is prime.

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A216835 Fibonacci + Goldbach (dual sequence to A216275). a(1)=5, a(2)=7 and for n>=3, a(n) = g(a(n-1) + a(n-2)), where for m>=3, g(2*m) is the maximal prime p < 2*m such that 2*m - p is prime.

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A129301 Records in A082467.

A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.

A216835 Fibonacci + Goldbach (dual sequence to A216275). a(1)=5, a(2)=7 and for n>=3, a(n) = g(a(n-1) + a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.