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User: Barry Cherkas

Barry Cherkas has authored 2 sequences.

A350713 Maximum smallest prime required to generate all Goldbach partitions to 10^n.

3, 19, 73, 173, 293, 523, 751, 1093, 1789, 1877, 2803, 3457, 3917, 4909, 5569, 6961, 7753, 9341

Offset: 1

Barry Cherkas, Feb 02 2022

The magnitude of the smallest prime required in a Goldbach partition of 2n is very small in comparison to the magnitude of the sum, 2n.

			The first three partitions with the smallest first member are (3,3), (3,5), and (3,7), so the smallest prime required to generate all Goldbach partitions up through 10^1 is 3.

Cf. A025018, A025019.

gp = Compile[{{n, Integer}}, Block[{p = 2}, While[! PrimeQ[n - p], p = NextPrime@p]; p]]; a[n] = 3; a[n] := Block[{k = 10^(n - 1), lmt = 10^n + 1, mx = 0}, While[k < lmt, b = gp@k; If[b > mx, mx = b]; k += 2]; mx]; (* Robert G. Wilson v, Mar 04 2022 *)

a(9)-a(18) from Robert G. Wilson v, Mar 04 2022

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).

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User: Barry Cherkas

A350713 Maximum smallest prime required to generate all Goldbach partitions to 10^n.

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

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