A298736 a(n) = s(n) - prime(n+1)+3, where s(n) = smallest even number x > prime(n) such that the difference x-p is composite for all primes p <= prime(n).

6, 10, 26, 90, 88, 84, 82, 200, 282, 280, 522, 518, 516, 512, 942, 936, 934, 928, 924, 922, 2566, 2562, 2556, 2548, 2544, 2542, 5268, 5266, 5262, 5248, 5244, 5238, 5236, 7280, 7278, 7272, 7266, 7262, 7256, 43356, 43354, 43344, 43342, 43338, 43336, 43324, 54024

Offset

1,1

Comments

The statement "a(n) >= 0 for n >= 1" is equivalent to Goldbach's conjecture (cf. Phong, Dongdong, 2004, Theorem (a)).

Records: 6, 10, 26, 90, 200, 282, 522, 942, 2566, 5268, 7280, 43356, 54024, ..., . - Robert G. Wilson v, Feb 28 2018

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..1205

Bui Minh Phong and Li Dongdong, Elementary problems which are equivalent to the Goldbach's Conjecture, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 33-37.

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

Formula

a(n) = A152522(n)-A000040(n+1)+3 for n > 0.

Maple

N:= 100: # to get a(1)..a(N)
P:= [seq(ithprime(i), i=1..N+1)]:
s:= proc(n, k0) local k;
  for k from max(k0, P[n]+1) by 2 do
    if andmap(not(isprime), map(t -> k - t, P[1..n])) then return k
  fi
od
end proc:
K[1]:= 6: A[1]:= 6:
for n from 2 to N do
  K[n]:= s(n, K[n-1]);
  A[n]:= K[n]- P[n+1]+3;
od:
seq(A[n], n=1..N); # Robert Israel, Mar 01 2018

Mathematica

f[n_] := Block[{k, x = 2, q = Prime@ Range@ n}, x += Mod[x, 2]; While[k = 1; While[k < n +1 && CompositeQ[x - q[[k]]], k++]; k < n +1, z = x += 2]; x - Prime[n +1] +3]; Array[f, 47] (* Robert G. Wilson v, Feb 26 2018 *)

Prog

(PARI) s(n) = my(p=prime(n), x); if(p==2, x=4, x=p+1); while(1, forprime(q=1, p, if(ispseudoprime(x-q), break, if(q==p, return(x)))); x=x+2)
a(n) = s(n)-prime(n+1)+3

Crossrefs

Cf. A000040, A082467, A152522, A242165.

Sequence in context: A351446 A323107 A077621 * A336845 A324617 A359145

Adjacent sequences: A298733 A298734 A298735 * A298737 A298738 A298739

Keyword

look,nonn

Author

Felix Fröhlich, Jan 25 2018

Status

approved