A002373 -id:A002373 - cp's OEIS frontend

A103153 a(n) is the smallest odd prime p such that 2n+1 = 2p + A000040(k) for some k>1, or 0 if no such prime exists.

0, 0, 0, 3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 5, 5, 3, 7, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 5, 5, 3, 7, 3, 3, 5, 5, 3, 7, 3, 19, 5, 3, 7, 5, 11, 3, 11, 3, 3, 5, 3, 3, 5, 3, 7, 5, 11, 7, 11, 11, 3, 11, 3, 13, 5, 3, 3, 5, 5, 7, 7, 3, 3, 5, 5, 3, 7, 5, 3, 7, 3, 13, 5, 3, 7, 5, 3, 3, 5, 5, 7, 7, 3

Offset: 1

Lei Zhou, Feb 09 2005

nonn

			For n < 4 there are no such primes, thus a(1)-a(3)=0. For n=4, 2*4+1 = 9 = 2*3+3, thus a(4)=3. For n=7, 2*7+1 = 15 = 2*5+5, thus a(7)=7.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

a(n)=0 if A103507(n)=0, otherwise A000040(A103507(n)).
Cf. A103151, A103152, A002373.
Cf. A195352 (similar definition, but p=2 is allowed).

Do[m = 3; While[ ! (PrimeQ[m] && ((n - 2*m) > 2) && PrimeQ[n - 2*m]), m = m + 2]; Print[m], {n, 9, 299, 2}]

(define (A103153 n) (let ((ind (A103507 n))) (if (zero? ind) 0 (A000040 ind))))

Edited and Scheme code added by Antti Karttunen, Jun 19 2007

Definition corrected by Hugo Pfoertner, Sep 16 2011

A049613 a(n) = 2n - (largest prime < 2n-2).

Original entry on oeis.org

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, 9, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 9, 11, 13, 15, 3, 5, 3, 5, 7, 3, 3, 5, 7, 9, 11, 3, 3, 5, 7, 3, 5, 7, 3, 5, 3, 5, 7, 3, 5, 7, 3, 3, 5

Offset: 3

David M. Elder (elddm(AT)rhodes.edu)

			a(14)=28 - (largest prime < 26) = 28 - 23 = 5.

T. D. Noe, Table of n, a(n) for n=3..10000

Cf. A049653, A049711, A049716, A049847.

Cf. A002373, A020481, A007917.

a049613 n = 2 * n - a007917 (2 * n - 2)
-- Reinhard Zumkeller, Jan 02 2015

Table[2n-NextPrime[2n-2,-1],{n,3,100}] (* Harvey P. Dale, Aug 16 2011 *)

a(n) <= A002373(n). - R. J. Mathar, Mar 19 2008

a(n) = 2*n - A007917(2*n-2). - Reinhard Zumkeller, Jan 02 2015

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

A002092 From a Goldbach conjecture: records in A185091.

Original entry on oeis.org

1, 3, 5, 7, 17, 29, 47, 61, 73, 83, 277, 317, 349, 419, 503, 601, 709, 829, 877, 1129, 1237, 1367, 1429, 1669, 1801, 2467, 2833, 2879, 3001, 3037, 3329, 3821, 4861, 5003, 5281, 5821, 5897, 6301, 6329, 6421, 6481, 6841, 7069, 7121, 7309, 7873, 8017, 8597, 8821

Offset: 1

N. J. A. Sloane

nonn

See A002091. The sequence gives the record values of q in the representations minimizing q of 2*k+1 = 2*p+q, p prime, q {1,prime}.

Checked up to 2*k = 10^13.

Brian H. Mayoh, On the second Goldbach conjecture. Nordisk Tidskr. Informations-Behandling 6, 1966, pp. 48-50.
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).

See A002091.
Index entries for sequences related to Goldbach conjecture

Cf. A002372, A002373, A002374, A002375, A045917, A006307, A002091, A185091, A194828, A194829.

a(n) = A185091((A002091(n)+1)/2).

Comment added, a(19)-a(32) from Hugo Pfoertner, Sep 03 2011
a(33) from Jason Kimberley, a(34)-a(40) from Hugo Pfoertner, Sep 09 2011
a(41)-a(49) from Hugo Pfoertner, Sep 25 2011

A051610 Monotonic subsequence of A051169.

Original entry on oeis.org

3, 6, 15, 49, 110, 154, 278, 496, 1321, 2686, 3713, 21766, 27122, 31637, 56836, 64084, 97214, 97235, 206786, 251611, 538711, 1763479, 1903702, 5379961, 12053441, 13894939, 18999469, 30059956, 56816411, 93926431, 167535419, 209955962, 360506719, 923566921

Offset: 1

Paul Bruckman (pbruckman(AT)hotmail.com)

nonn

Indices of records of A002373. - Ralf Stephan, Dec 29 2013

Cf. A051169, A116111.

More terms from Paul S. Bruckman, Jan 20 2007

Edited by N. J. A. Sloane, Apr 15 2007, May 04 2007, Jun 10 2008

A063713 Numbers n such that there exist primes p, q, r with n*2 = p - r = r + q (values of r are given in A063714).

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 27, 28, 30, 32, 33, 35, 36, 38, 39, 42, 43, 45, 46, 48, 50, 51, 52, 53, 54, 55, 57, 58, 60, 63, 65, 66, 67, 69, 70, 71, 72, 75, 77, 78, 80, 81, 84, 85, 87, 88, 90, 93, 96, 97, 98, 99, 100, 101, 102, 105

Offset: 1

Reinhard Zumkeller, Aug 10 2001

nonn

			10*2 = 20 = 23 - 3 = 3 + 17, A063714(7) = 3; 11*2 = 22 = 41 - 19 = 19 + 3, A063714(8) = 19 28 is missing because we have the prime sums (Goldbach): 5 + 23 = 11 + 17 and differences with primes less 28: 31 - 3 = 41 - 13 = 47 - 19; none of these have a prime in common.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A063714, A002373, A020483.

filter:= proc(n) local k;
  k:= 1;
  while k < 2*n do
    k:= nextprime(k);
    if isprime(2*n+k) and isprime(2*n-k) then return true fi
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Oct 09 2017

okQ[n_] := AnyTrue[Prime[Range[PrimePi[2 n - 2]]], PrimeQ[2 n + #] && PrimeQ[2 n - #]&]; Select[Range[105], okQ] (* Jean-François Alcover, Feb 12 2018 *)

A208662 Smallest m such that the n-th odd prime is the smallest prime for all decompositions of 2*m into two primes.

Original entry on oeis.org

3, 6, 15, 62, 61, 209, 49, 110, 173, 154, 637, 572, 481, 278, 1256, 1763, 691, 928, 2309, 496, 1909, 3716, 6389, 2989, 13049, 1321, 11633, 5134, 9848, 3004, 17096, 11303, 2686, 18884, 6781, 4798, 11416, 29957, 3713, 44393, 25156, 48884, 24001, 56279, 30031

Offset: 1

Reinhard Zumkeller, Feb 29 2012

nonn

A002373(a(n)) = A065091(n) and A002373(m) != A065091(n) for m < a(n).

			n=3, a(3)=15: 7 is the 3rd odd prime and the smallest prime in all Goldbach decompositions of 2*15 = 30 = {7+23, 11+19, 13+17}, and 7 doesn't occur as smallest prime in all Goldbach decompositions for even numbers less than 30.

Reinhard Zumkeller, Table of n, a(n) for n = 1..120
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A002373, A065091, A260485.

a208662 n = head [m | m <- [1..], let p = a065091 n,
   let q = 2 * m - p, a010051' q == 1,
   all ((== 0) . a010051') $ map (2 * m -) $ take (n - 1) a065091_list]
-- Reinhard Zumkeller, Aug 11 2015, Feb 29 2012

A063714 Values of r occurring in A063713.

Original entry on oeis.org

3, 3, 5, 3, 3, 5, 3, 19, 5, 3, 7, 29, 3, 5, 3, 5, 3, 43, 5, 3, 7, 3, 7, 3, 5, 3, 11, 3, 5, 5, 3, 7, 89, 7, 3, 5, 3, 3, 5, 3, 13, 113, 7, 13, 127, 5, 3, 11, 137, 139, 5, 13, 3, 7, 3, 5, 5, 3, 7, 3, 13, 5, 19, 3, 3, 31, 197, 199, 7, 13, 17, 11, 3, 5, 3, 229, 5, 3, 11, 5, 11, 3, 19, 3, 7, 3, 7

Offset: 1

Reinhard Zumkeller, Aug 10 2001

This is not a mere union of A002373 and A020483 because of the minimality property of these sequences.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A063713, A002373, A020483.

f:= proc(n) local k;
k:= 2;
  while k < 2*n do
   k:= nextprime(k);
    if isprime(2*n+k) and isprime(2*n-k) then return k fi
  od;
  NULL
end proc:
map(f, [$1..400]); # Robert Israel, Oct 09 2017

f[n_] := {AnyTrue[Prime[Range[PrimePi[2n-2]]], (r = #; PrimeQ[2n+r] && PrimeQ[2n-r])&], r}; Select[f /@ Range[200], #[[1]] =!= False &][[All, 2]] (* Jean-François Alcover, Feb 14 2018 *)

A064466 a(0) = 6 and a(n) = Min { m > a(n-1) | both a(n-1) - p and m - p are prime for some prime p } for n > 0.

Original entry on oeis.org

6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 72, 74, 76, 78, 80, 84, 86, 88, 90, 92, 94, 96, 98, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 126, 128, 132, 134, 136, 138, 140, 142

Offset: 0

Reinhard Zumkeller, Oct 02 2001

nonn

The initially very frequent case a(k+1) = a(k) + 2 means that there is a twin prime (q, q + 2) with a(k+1) = p + (q + 2) and a(k) = p + q. This might illustrate a certain coherence of two famous conjectures: Goldbach and twin primes.

			a(12) = 30 = 13 + 17: a(13) = 30 + 2 = 32 = 13 + 19 (common prime = 13). No common prime exists in Goldbach decompositions for a(16) = 38 and 40, so 40 <> a(17) = 42; a(16) = 38 = 7 + 31 = 19 + 19, 40 = 3 + 37 = 11 + 29 = 17 + 23, a(17) = 42 = 11 + 31 (for 38 and 42 common prime = 31); A064634(1) = 16, A064635(1) = 40 = 2 + 38 = 2 + a(A064634(1)).

Cf. A001359, A002373, A064634, A064635.

A235649 Define a(4)=3 then a(n+1)is the smallest prime P such that a(n)<=P< n with 2*n-P=Q prime and if not possible a(n+1) is the smallest prime P such that P

Original entry on oeis.org

3, 3, 5, 3, 3, 5, 7, 3, 5, 7, 11, 11, 13, 3, 5, 7, 11, 11, 13, 17, 17, 19, 23, 23, 3, 5, 7, 19, 23, 23, 31, 3, 5, 7, 17, 17, 19, 23, 23, 3, 5, 7, 13, 23, 23, 31, 41, 41, 43, 47, 47, 3, 3, 5, 7, 11, 11, 13, 17, 17, 19, 23, 23, 31, 47, 59, 61, 3, 5, 7, 11

Offset: 4

Pierre CAMI, Jan 13 2014

nonn

			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)=3 as 2*7-3=11 prime
a(8)=3 as 2*8-3=13 prime

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

Cf. A002373, A020483

cp's OEIS Frontend

A103153 a(n) is the smallest odd prime p such that 2*n+1 = 2*p + A000040(k) for some k>1, or 0 if no such prime exists.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Extensions

A049613 a(n) = 2n - (largest prime < 2n-2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A002092 From a Goldbach conjecture: records in A185091.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A051610 Monotonic subsequence of A051169.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A063713 Numbers n such that there exist primes p, q, r with n*2 = p - r = r + q (values of r are given in A063714).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A208662 Smallest m such that the n-th odd prime is the smallest prime for all decompositions of 2*m into two primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

A063714 Values of r occurring in A063713.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A103153 a(n) is the smallest odd prime p such that 2n+1 = 2p + A000040(k) for some k>1, or 0 if no such prime exists.