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A298736
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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).
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1
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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