A007534 -id:A007534 - cp's OEIS frontend

A072254 Numbers n such that triples generated by {2(n-1),2n,2*(n+1)} form A007534.

1, 48, 201, 258, 393, 453, 558, 573, 633, 678, 1623, 2103

Offset: 1

Lekraj Beedassy, Jul 08 2002

nonn

A129363 Number of partitions of 2n into the sum of two twin primes.

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 1, 2, 2, 3, 3, 2, 1, 2, 2, 3, 4, 2, 1, 2, 1, 2, 3, 3, 2, 2, 1, 2, 4, 3, 3, 4, 2, 2, 3, 2, 2, 4, 2, 0, 0, 0, 2, 4, 3, 2, 2, 2, 4, 6, 3, 3, 5, 3, 1, 2, 1, 2, 4, 2, 1, 2, 2, 4, 5, 3, 2, 4, 3, 3, 4, 2, 2, 4, 2, 3, 6, 3, 1, 2, 1, 3, 6, 4, 2, 2, 1, 2, 4, 3, 4, 6, 4, 4, 5, 3, 6, 12

Offset: 1

T. D. Noe, Apr 11 2007

nonn

a(n/2)=0 for the n in A007534. The logarithmic plot of this sequence seems very regular after 200000 terms.

			a(11)=3 because 22 = 3+19 = 5+17 = 11+11.

T. D. Noe, Table of n, a(n) for n=1..10000
James Grime and Brady Haran, Goldbach Conjecture (but with TWIN PRIMES), Numberphile video (2024)
T. D. Noe, Logarithmic plot of 10^6 terms

Cf. A175931 (n for which a(n-1), a(n), a(n+1) are equal).

Cf. A001097, A002375, A007534.

a129363 n = sum $ map (a164292 . (2*n -)) $ takeWhile (<= n) a001097_list
-- Reinhard Zumkeller, Feb 03 2014

nn=1000; tw=Select[Prime[Range[PrimePi[nn]]], PrimeQ[ #+2]&]; tw=Union[tw,tw+2]; tc=Table[0,{nn}]; tc[[tw]]=1; Table[cnt=0; k=1; While[tw[[k]]<=n/2, cnt=cnt+tc[[n-tw[[k]]]]; k++ ]; cnt, {n,2,nn,2}]

a(n) = Sum_{i=1..n} ceiling((A010051(i+2) + A010051(i-2))/2) * ceiling((A010051(2n-i+2) + A010051(2n-i-2))/2) * A010051(2n-i) * A010051(i). - Wesley Ivan Hurt, Jan 30 2014

a(n) = sum(A164292(2*n - A001097(k)): A001097(k) <= n). - Reinhard Zumkeller, Feb 03 2014

Comment converted to crossref by Klaus Brockhaus, Oct 27 2010

A113275 Lesser of twin primes for which the gap before the following twin primes is a record.

Original entry on oeis.org

3, 5, 17, 41, 71, 311, 347, 659, 2381, 5879, 13397, 18539, 24419, 62297, 187907, 687521, 688451, 850349, 2868959, 4869911, 9923987, 14656517, 17382479, 30752231, 32822369, 96894041, 136283429, 234966929, 248641037, 255949949

Offset: 1

Bernardo Boncompagni, Oct 21 2005

nonn

			The smallest twin prime pair is 3, 5, then 5, 7 so a(1) = 3; the following pair is 11, 13 so a(2) = 5 because 11 - 5 = 6 > 5 - 3 = 2; the following pair is 17, 19: since 17 - 11 = 6 = 11 - 5 nothing happens; the following pair is 29, 31 so a(3)= 17 because 29 - 17 = 12 > 11 - 5 = 6.

Martin Raab, Table of n, a(n) for n = 1..82 (first 75 terms from Max Alekseyev)
Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013.
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Mersenneforum, Gaps between prime pairs (Twin Primes).
Tomás Oliveira e Silva, Gaps between twin primes

Record gaps are given in A113274. Cf. A002386.

NextLowerTwinPrim[n_] := Block[{k = n + 2}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k++ ]; k]; p = 3; r = 0; t = {3}; Do[q = NextLowerTwinPrim[p]; If[q > r + p, AppendTo[t, p]; r = q - p]; p = q, {n, 10^9}] (* Robert G. Wilson v, Oct 22 2005 *)

a(n) = A036061(n) - 2.

a(n) = A036062(n) - A113274(n).

a(22)-a(30) from Robert G. Wilson v, Oct 22 2005

Terms up to a(72) are listed in Kourbatov (2013), terms up to a(75) in Oliveira e Silva's website, added by Max Alekseyev, Nov 06 2015

A266948 Least prime p such that p-2 and 6n-p are also prime, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 5, 5, 5, 7, 5, 5, 5, 7, 7, 5, 5, 5, 5, 7, 7, 5, 5, 5, 7, 13, 5, 7, 5, 13, 5, 5, 5, 7, 7, 5, 13, 5, 5, 13, 5, 31, 5, 5, 7, 5, 13, 7, 7, 7, 5, 5, 5, 13, 7, 13, 5, 5, 7, 13, 5, 5, 31, 5, 7, 7, 5, 5, 5, 7, 7, 5, 7, 5, 19, 5, 13, 5, 5, 7, 7, 5, 5, 7, 13, 7, 5, 7, 5, 7, 7, 13, 5, 13, 19, 5, 5, 109, 7, 7, 5, 5, 19, 7, 7, 5, 5, 5, 5, 13, 5, 43, 5, 7, 7, 5, 13, 5, 7, 7, 5, 19, 7, 5, 19

Offset: 0

M. F. Hasler, Jan 06 2016

nonn

Goldbach conjecture related: Group the consecutive even numbers in groups of three, (6n-2, 6n, 6n+2). The existence of a(n) corresponds to a Goldbach decomposition 6n = p + (6n-p) using the upper of a twin prime pair. Then 6n-2 = (p-2) + 6n-p is automatically a valid Goldbach decomposition of 6n-2, and 6n+2 = p + 6n+2-p is such a decomposition for 6n+2 if 6n+2-p (or 6n+4-p) is prime.
Zwillinger conjectured already in 1978 that for all n > 701 there is a p such that all these conditions are satisfied (not necessarily p = a(n)). See also A266952 - A266953.
This conjecture implies that a(n) > 0 for all n > 1.
See A266950 - A266951 for record values and indices. For easier reference we list some of these [n, a(n)] here: [21, 13]; [133, 139]; [1759, 241]; [10919, 643],  [112723, 1621]; [1072318, 2311], [1458993, 3001], [2617393, 3301], ...
Since a larger value of a(n) indicates that it was "difficult" to find a suitable twin prime p, this slow growth is a strong evidence that a(n) > 0 for all n > 1.

Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
Bill Krys, Not much response but I still think this is outrageous result, Yahoo! group primenumbers, Jan. 6, 2016
Bill Krys, not much response, but i still think this is outrageous result, message in primenumbers Yahoo group, Jan 6, 2016 [cached copy].
Dan Zwillinger, A Goldbach Conjecture Using Twin Primes, Math. Comp. 33, No.147 (1979), p.1071.

A266948(n)=my(GP(n,p=2)=forprime(p=p,n,isprime(n*2-p)&&return(p)));for(p=1,3*n,isprime(-2+p=GP(3*n,p))&&return(p))

A266952 Least prime p such that p-2 and 6n-p and 6n+2-p are also prime, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 7, 7, 7, 13, 7, 13, 7, 13, 19, 7, 13, 7, 13, 19, 0, 31, 7, 7, 13, 19, 31, 31, 7, 13, 7, 13, 19, 73, 31, 7, 13, 7, 7, 13, 19, 31, 31, 7, 13, 7, 13, 19, 73, 31, 7, 13, 7, 13, 19, 109, 31, 7, 13, 19, 109, 31, 109, 7, 13, 19, 61, 31, 73, 43, 199, 0, 61, 103, 73, 7, 13, 7, 13, 19, 109, 31, 7, 13, 19, 139, 31, 151, 43, 199, 0, 61, 7, 13, 19, 199, 31, 139, 43

Offset: 0

M. F. Hasler, Jan 06 2016

nonn

If a(n) > 0, then the triple {6n-2, 6n, 6n+2} of consecutive even numbers allows a "simultaneous Goldbach decomposition" using two pairs of twin primes, 6n-2 = p-2 + 6n-p ; 6n = p + 6n-p ; 6n+2 = p + 6n+2-p.
Up to 10^5, the only indices for which a(n)=0 are {0, 1, 16, 67, 86, 131, 151, 186, 191, 211, 226, 541, 701}. I conjecture that this list is finite, and probably complete. Is it a coincidence that all odd numbers > 1 in this list are primes? (See also A144094.)
This seems equivalent to a conjecture Zwillinger made in 1978, see reference in LINKS.
See A266953 for another variant with a slightly relaxed condition (instead of 6n+2-p one can also have 6n+4-p prime, but this affects only n=2 and n=67), and A266948 for another variant with less restrictive conditions (only p-2 and 6n-p have to be prime).

Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
Dan Zwillinger, A Goldbach Conjecture Using Twin Primes, Math. Comp. 33, No.147 (1979), p.1071.

Cf. A007534, A266948, A266953.

A266952(n)=my(GP(n, p=2)=forprime(p=p, n+1, isprime(n*2-p)&&return(p))); for(p=1, 3*n, isprime(-2+p=GP(3*n, p))+!p&&(!p||isprime(6*n+2-p))&&return(p))

A212613 Conjectured largest number having n representations as the sum of two twin primes p + q with p >= q.

Original entry on oeis.org

4208, 24536, 28916, 21278, 51806, 68078, 73538, 89216, 83978, 171146, 111116, 130586, 164876, 171788, 128498, 188546, 188816, 205796, 247376, 258968, 254546, 287306, 275456, 323966, 351788, 302816, 325148, 368378, 445688, 380168, 467696, 455486, 553118, 456608

Offset: 0

T. D. Noe, May 30 2012

nonn

These numbers were found by looking at all sums <= 5000000 of two twin primes.

			a(0) = 4208 is the last term of A007534.

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A212888 (number of such representations).

A064409 Positive even numbers not of the form A001359(i) + A006512(j) for integers i and j.

Original entry on oeis.org

2, 4, 6, 14, 20, 26, 28, 32, 38, 40, 44, 50, 52, 56, 58, 62, 68, 70, 74, 80, 82, 86, 88, 92, 94, 96, 98, 100, 104, 110, 116, 118, 122, 124, 128, 130, 134, 136, 140, 146, 148, 152, 158, 160, 164, 166, 170, 172, 176, 178, 182, 188, 190, 194, 200

Offset: 1

Robert G. Wilson v, Sep 29 2001

nonn

This is different from A007534, which gives positive even numbers which are not the same of a pair of twin primes. The old definition of the present sequence was misleading. - N. J. A. Sloane, Feb 16 2024

			The lesser of the twin primes < 200 are 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197 and the greater of the twin primes < 200 are 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199. 20 is in the current sequence because no combination of any two numbers from each set just enumerated can be summed to make 20.

A.H.M. Smeets, Table of n, a(n) for n = 1..20177

Cf. A001359, A006512, A007534.

p = Select[ Range[ 200 ], PrimeQ[ # ] && PrimeQ[ # + 2 ] & ]; q = p + 2; Complement[ Table[ n, {n, 2, 200, 2} ], Union[ Flatten[ Table[ p[ [ i ] ] + q[ [ j ] ], {i, 1, 15}, {j, 1, 15} ] ] ] ]

A212888 Conjectured number of positive even numbers that can be represented as the sum of two twin primes p + q in exactly n ways, with p >= q.

Original entry on oeis.org

35, 115, 285, 327, 557, 537, 723, 652, 882, 773, 1061, 916, 1158, 1069, 1369, 1143, 1436, 1307, 1566, 1318, 1711, 1336, 1752, 1380, 1798, 1480, 1847, 1449, 2012, 1556, 1967, 1599, 2184, 1646, 2257, 1656, 2278, 1779, 2376, 1786, 2353, 1823, 2523, 1832, 2415

Offset: 0

Michael Jones, May 29 2012

nonn

For even n, a(n) - a(n-1) increases as n increases.

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A001097 (twin primes).
Cf. A007534 (35 numbers that have no representation).
Cf. A129363 (number of partitions of 2n into the sum of two twin primes).
Cf. A212613 (last number having n representations).

p = Prime[Range[10000]]; tp = Select[p, PrimeQ[# + 2] || PrimeQ[# - 2] &]; {tn, t} = Transpose[Sort[Tally[Select[Flatten[Table[If[a >= b, a + b, 0], {a, tp}, {b, tp}]], # < tp[[-1]] &]]]]; Join[{Length[Complement[Range[2, tn[[-1]], 2], tn]]}, Table[Count[t, n], {n, 8}]] (* T. D. Noe, May 30 2012 *)

Extended and corrected by T. D. Noe, May 30 2012

A051345 Odd numbers not of the form 3 + twin prime + twin prime.

Original entry on oeis.org

1, 3, 5, 7, 97, 99, 101, 403, 405, 407, 517, 519, 521, 787, 789, 791, 907, 909, 911, 1117, 1119, 1121, 1147, 1149, 1151, 1267, 1269, 1271, 1357, 1359, 1361, 3247, 3249, 3251, 4207, 4209, 4211

Offset: 1

David W. Wilson

Index entries for sequences related to Goldbach conjecture

Essentially A007534 + 3.

tps=Flatten[Select[Partition[Prime[Range[2000]],2,1], Last[#]-First[#]==2&]]; Complement[Range[1,Last[tps],2], Union[Total[#]+3&/@Tuples[tps,{2}]]]  (* Harvey P. Dale, Apr 01 2011 *)

No others below 100000. Conjectured to be complete.

A144094 Prime numbers r such that there is no pair of twin primes p,p+2 and q,q+2 such that p+q+2 = 6r.

Original entry on oeis.org

67, 131, 151, 191, 211, 541, 701

Offset: 1

Pierre CAMI, Sep 10 2008

nonn

Probably the sequence is complete.

More generally, {1, 2, 16, 66, 67, 86, 116, 131, 151, 186, 191, 211, 226, 541, 701} seem to be the only numbers such that {6n-2, 6n, 6n+2} do not have a Goldbach type of decomposition (sum of two primes) using only two pairs of twin primes. See also A266952, A266953, and A266948 and A007534. - M. F. Hasler, Jan 07 2016

			6*3=18, 5,7 twin primes as 11,13, 5+13=7+11=18 so 3 not in the sequence 6*5=30, 11,13 twin primes as 17,19, 13+19=11+17=30 so 5 not in the sequence

cp's OEIS Frontend

A072254 Numbers n such that triples generated by {2*(n-1),2*n,2*(n+1)} form A007534.

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A129363 Number of partitions of 2n into the sum of two twin primes.

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A113275 Lesser of twin primes for which the gap before the following twin primes is a record.

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A266948 Least prime p such that p-2 and 6n-p are also prime, or 0 if no such prime exists.

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A266952 Least prime p such that p-2 and 6n-p and 6n+2-p are also prime, or 0 if no such prime exists.

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A212613 Conjectured largest number having n representations as the sum of two twin primes p + q with p >= q.

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A064409 Positive even numbers not of the form A001359(i) + A006512(j) for integers i and j.

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A212888 Conjectured number of positive even numbers that can be represented as the sum of two twin primes p + q in exactly n ways, with p >= q.

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A051345 Odd numbers not of the form 3 + twin prime + twin prime.

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A072254 Numbers n such that triples generated by {2(n-1),2n,2*(n+1)} form A007534.