A139602 -id:A139602 - cp's OEIS frontend

A189888 a(n) = A139602(m) such that for any k>m, A139602(k) > A139602(m).

11, 19, 43, 97, 163, 191, 223, 457, 877, 1049, 1307, 1987, 2029, 2129, 4217, 6599, 9967, 10357, 18233, 66343, 74573, 95911, 99719, 186551, 196337, 211219, 262469, 277301, 416573, 603487, 994549, 1403137, 4117441, 4805761, 4895789, 5823067, 5842813, 7704409

Offset: 1

Lei Zhou, Apr 30 2011

nonn

Dickson's conjecture implies that this sequence is infinite. - Charles R Greathouse IV, Mar 22 2011

			For n=1, a(n)=A139602(1)=11;
A139602(3)=61 is skipped since A139602(4)=43 < A139602(3), thus a(3)=A139602(4)=43;
For the same reason, A139602(8) and A139602(9) are skipped so that a(7)=A139602(10)=223.

Lei Zhou, List of prime pair a(n) +/- 6*k.

Cf. A139602.

count=0; imin=0; cstop=65; pn=4; While[countimin,imin=i; count++]]

A268914 Minimum difference between two distinct primes whose sum is 2*prime(n), n>4.

Original entry on oeis.org

12, 12, 12, 24, 12, 24, 24, 12, 24, 48, 12, 12, 24, 36, 12, 24, 12, 36, 48, 36, 60, 24, 12, 12, 60, 48, 48, 36, 60, 24, 36, 24, 12, 72, 60, 12, 24, 36, 84, 60, 60, 84, 24, 120, 60, 96, 12, 24, 60, 24, 12, 12, 24, 84, 12, 24, 108, 48, 48, 84, 72, 72, 36, 60, 72, 36, 12, 84, 60, 12, 60, 72, 60, 48, 36, 24, 60, 24, 24, 48, 36, 48, 36, 168, 36, 48

Offset: 5

Barry Cherkas, Feb 15 2016

If p>4 is prime, any two primes that add to 2p must be equidistant from p. If p is congruent to 1 Mod 3, then p+2 and p-4 are divisible by 3. Alternatively, if p is congruent to 2 Mod 3, the p-2 and p+4 are divisible by 3. Thus, the equidistant pairs (p-2,p+2) and (p-4,p+4) cannot be primes that add to 2p. On the other hand, adding or subtracting any multiple of 6 will be congruent to the same congruence class as p and may be prime. Thus, the minimal difference between distinct primes that add to p must be a multiple of 12.

Extrapolating from computational evidence for all primes up to 10^9, we conjecture: For each multiple of 12 there are infinitely many primes p such that p-6k and p+6k are prime and 12k is the minimal difference for two distinct primes whose sum is 2p.

			For n=5, 2*prime(5)=2*11=5+17 and 17-5=12.
For n=6, 2*prime(6)=2*13=7+19 and 19-7=12.
...
For n=8, 2*prime(8)=2*19=7+31 and 31-7=24.

Barry Cherkas, Table of n, a(n) for n = 5..10003
G. H. Hardy and J. E. Littlewood, Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes, Acta Math. 44, 1-70, 1923.

Cf. A078611, A071681, A139602, A006489, A137796, A161723, A128940.

N:= 1000: # to get a(5) .. a(N)
p:= 7:
for n from 5 to N do
  p:= nextprime(p);
  for k from 6 by 6 while not isprime(p+k) or not isprime(p-k) do od:
  A[n]:= 2*k
od:
seq(A[n],n=5..N); # Robert Israel, Mar 09 2016

f[n_]:=Block[{p=Prime[n],k},k=p+6;
While[!PrimeQ[k]||!PrimeQ[2p-k],k=k+6];2(k-p)];
seq=Reap[Do[Sow[f[n]],{n,5,200}]][[2]][[1]];
seq
(*For large data sets (say, N>5000), replace 200 with N and the above algorithm is comparatively efficient.*)
Table[2 SelectFirst[Range[#/2], Function[k, AllTrue[{#/2 + k, #/2 - k}, PrimeQ]]] &[2 Prime@ n], {n, 5, 120}] (* Michael De Vlieger, Mar 09 2016, Version 10 *)

a(n) = {p = prime(n); d = 2; while (! (isprime(p-d) && isprime(p+d)),  d+=2); 2*d;} \\ Michel Marcus, Mar 17 2016

a(n) = 2*A078611(n+2).

cp's OEIS Frontend

A189888 a(n) = A139602(m) such that for any k>m, A139602(k) > A139602(m).

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