OFFSET
1,5
COMMENTS
It is hypothesized that all terms of this sequence are positive integers.
If the above hypothesis is true, the Goldbach Hypothesis is true, since for every even number 2n, if there is a Goldbach decomposition p+q=2n meets the condition of this sequence, p+q+6=2n+6 forms at least one Goldbach decomposition of 2n+6.
EXAMPLE
For n=1, 2*(n+4)=10, 10=5+5, and 5+6=11 is a prime. Thus a(1)=1;
For n=2, 2*(n+4)=12, 12=5+7, and 5+6=11 is a prime. Thus a(2)=1;
...
For n=14, 2*(n+4)=36, 36=5+31 (5+6=11); 7+29 (7+6=13); 13+23 (13+6=19); 17+19 (17+6=23), four cases found. Thus a(14)=4.
MATHEMATICA
res = {}; Do[n[2] = i*6; n[1] = n[2] - 2; n[3] = n[2] + 2;
Do[c[j] = 0; p[j] = NextPrime[n[j]/2 - 1];
While[q[j] = n[j] - p[j];
If[PrimeQ[q[j]] && q[j] > 3,
If[PrimeQ[p[j] + 6] || PrimeQ[q[j] + 6], c[j]++]];
p[j] < n[j] - 5, p[j] = NextPrime[p[j]]], {j, 1, 3}];
AppendTo[res, c[1]]; AppendTo[res, c[2]];
AppendTo[res, c[3]], {i, 2, 29}]; Print[res]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lei Zhou, Sep 25 2024
STATUS
approved