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A235330 Number of ways to write 2*n = p + q with p, q, prime(p) - p + 1 and prime(q) + q + 1 all prime. 5
0, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 3, 1, 1, 2, 3, 0, 1, 2, 0, 3, 1, 0, 2, 2, 0, 0, 1, 1, 2, 3, 0, 1, 3, 0, 2, 0, 0, 2, 1, 0, 1, 2, 0, 3, 0, 0, 4, 2, 1, 1, 1, 1, 3, 4, 1, 1, 3, 1, 0, 2, 1, 1, 3, 0, 0, 2, 3, 3, 3, 1, 1, 3, 3, 2, 3, 1, 1, 5, 0, 1, 4, 2, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

Conjecture: (i) a(n) > 0 for all n >= 2480.

(ii) If n > 4368 then 2*n+1 can be written as 2*p + q with p and q terms of the sequence A234695.

Parts (i) and (ii) are stronger than Goldbach's conjecture  (A045917) and Lemoine's conjecture (A046927) respectively.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

EXAMPLE

a(8) = 1 since 2*8 = 5 + 11 with 5, 11, prime(5) - 5 + 1 = 7 and prime(11) + 11 + 1 = 43 all prime.

MATHEMATICA

p[n_] := PrimeQ[n] && PrimeQ[Prime[n] - n + 1];

q[n_] := PrimeQ[n] && PrimeQ[Prime[n] + n + 1];

a[n_] := Sum[If[p[k] && q[2 n - k], 1, 0], {k, 1, 2 n - 1}];

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000040, A045917, A046927, A234695, A235187, A235189.

Sequence in context: A215401 A254606 A175358 * A029394 A035467 A254045

Adjacent sequences:  A235327 A235328 A235329 * A235331 A235332 A235333

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 05 2014

STATUS

approved

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Last modified July 5 00:40 EDT 2020. Contains 335457 sequences. (Running on oeis4.)