A066286 -id:A066286 - cp's OEIS frontend

A065978 For even k >= 4, let f(k) = A066285(k/2) be the minimal difference between primes p and q whose sum is k. Such a k is in the sequence if f(k) > f(m) for all even m with 4 <= m < k.

4, 8, 16, 44, 92, 242, 256, 272, 292, 476, 530, 572, 682, 688, 1052, 1808, 2228, 3382, 3472, 3502, 3562, 4952, 6194, 7102, 10262, 17008, 20684, 37052, 45128, 49552, 80144, 137414, 251806, 349826, 362534, 742856, 1655152, 1872236, 2108282, 2319728, 2707118

Offset: 1

Jon Perry, Dec 09 2001

The values of f(a(n)) (given in A066286) appear to be divisible by 6, except the first two.

			4 = 2+2; the gap is 0. 6=3+3 (0). 8=3+5; the gap is 2, and this is the largest gap to date, so 8 is in the sequence.
10=5+5 (0), 12=5+7 (2), 14=7+7 (0), 16=5+11 (6), so 16 is in the sequence.

Gilmar Rodriguez Pierluissi, Table of n, a(n) for n = 1..64 (terms 1..50 from Jon Perry, Robert G. Wilson and Dean Hickerson, terms 51..55 from Gilmar Rodriguez Pierluissi, terms 56..63 from Robert G. Wilson v)

Cf. A066285, A066286, A107926.

f[n_] := For[p=n/2, True, p--, If[PrimeQ[p]&&PrimeQ[n-p], Return[n-2p]]]; For[n=4; max=-1, True, n+=2, If[f[n]>max, Print[n]; max=f[n]]]

More terms from Robert G. Wilson v and Dean Hickerson, Dec 10 2001

Changed offset to 1 (this is a list). - N. J. A. Sloane, Sep 07 2013

A066285 a(n) is the minimal difference between primes p and q whose sum is 2n.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 6, 4, 6, 0, 2, 0, 6, 4, 6, 0, 2, 0, 6, 4, 18, 0, 10, 12, 6, 8, 18, 0, 2, 0, 18, 8, 6, 12, 10, 0, 18, 4, 6, 0, 2, 0, 6, 4, 30, 0, 10, 24, 6, 16, 18, 0, 14, 24, 6, 8, 30, 0, 2, 0, 18, 8, 6, 12, 10, 0, 30, 4, 6, 0, 2, 0, 30, 8, 6, 12, 10, 0, 18, 4, 30, 0, 10, 24, 6, 28, 18, 0

Offset: 2

Dean Hickerson, Dec 12 2001

Terms are always even numbers because primes present in Goldbach partitions of n > 4 are odd and n = 4 has just one partition (2+2) where the difference is 0. a(n) = 0 iff n is prime. - Marcin Barylski, Apr 28 2018

Cf. A002372, A047160, A065978, A066286, A303603.

a[n_] := For[p=n, True, p--, If[PrimeQ[p]&&PrimeQ[2n-p], Return[2n-2p]]]

a(n) = {forstep(k=n, 1, -1, if (isprime(k) && isprime(2*n-k), return(2*n-2*k)););} \\ Michel Marcus, Jun 01 2020

a(n) = 2 * A047160(n). - Alois P. Heinz, Jun 01 2020

A107926 The least number k such that there are primes p and q with p - q = 2*n, p + q = k, and p the least such prime >= k/2.

Original entry on oeis.org

4, 8, 18, 16, 54, 48, 50, 108, 102, 44, 234, 444, 98, 228, 174, 92, 414, 432, 242, 516, 582, 256, 1182, 672, 406, 612, 846, 272, 1038, 984, 442, 1776, 1902, 292, 1074, 636, 1054, 3312, 1122, 476, 1398, 1464, 530, 1728, 2730, 572, 2706, 3348, 682, 2844, 3342

Offset: 0

Gilmar J. Rodriguez (Gilmar.Rodriguez(AT)nwfwmd.state.fl.us) and Robert G. Wilson v, Jun 16 2005

nonn

From the Goldbach conjecture.
A107926 = 2*A103147 by definition.
a(3n)> a(3n-2), a(3n-1), a(3n+1) & a(3n+2) for all n > 0 except for n = 1, 2, 12, 19, 20 or 41.
Of those values found so far a(3n+2) > a(3n+1) by ~8%. - Robert G. Wilson v, Nov 03 2013
Except for 1, all indices, i, not congruent to 0 (mod 3), a(i) is congruent to 0 (mod 6) and for all indices, i, congruent to 0 (mod 3), a(i) is not congruent to 0 (mod 6). Of those not congruent to 0 (mod 6), those congruent to 2 outnumber those congruent to 4, about 8 to 7. Robert G. Wilson v, Nov 03 2013

			a(0) = 4 because 4=2+2 and 2-2=0.
a(1) = 8 because 8 is the least number with 8=p+q and p-q=2 for primes p and q.
a(2) = 18 because 18=7+11 and the primes 7 and 11 have difference 4.

Robert G. Wilson v, Table of n, a(n) for n = 0..2560 (first 501 terms from T. D. Noe)
Mark Herkommer, Goldbach Conjecture Research.
Tomás Oliveira e Silva, Goldbach conjecture verification.
The Prime Glossary, Goldbach's conjecture
+Plus Magazine ... living mathematics, Mathematical mysteries: the Goldbach conjecture
Eric Weisstein's World of Mathematics, Goldbach Conjecture
Wikipedia, Goldbach conjecture

Cf. A066285, A103147, records in A065978 and A066286.

f[n_] := For[p = n/2, True, p--, If[PrimeQ[p] && PrimeQ[n - p], Return[n/2 - p]]]; nn=101; t=Table[0,{nn}]; cnt=0; n=1; While[cnt

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A065978 For even k >= 4, let f(k) = A066285(k/2) be the minimal difference between primes p and q whose sum is k. Such a k is in the sequence if f(k) > f(m) for all even m with 4 <= m < k.

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A066285 a(n) is the minimal difference between primes p and q whose sum is 2n.

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A107926 The least number k such that there are primes p and q with p - q = 2*n, p + q = k, and p the least such prime >= k/2.

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