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A235189 Number of ways to write n = (1 + (n mod 2))*p + q with p < n/2 such that p, q and prime(p) - p + 1 are all prime. 9

%I

%S 0,0,0,0,0,0,1,1,2,1,2,1,2,1,2,2,4,2,2,2,3,2,3,2,2,2,3,1,3,2,2,2,4,2,

%T 2,4,4,1,3,2,3,2,3,3,4,3,4,3,3,3,5,2,4,4,2,2,6,2,2,4,1,1,5,4,5,4,4,2,

%U 4,3,3,3,4,4,5,4,5,4,3,2,4,2,3,6,5,3,6,3,5,5,2,3,9,3,3,5,3,1,6,3

%N Number of ways to write n = (1 + (n mod 2))*p + q with p < n/2 such that p, q and prime(p) - p + 1 are all prime.

%C Conjecture: a(n) > 0 for all n > 6.

%C This implies both Goldbach's conjecture (A045917) and Lemoine's conjecture (A046927). For primes p with prime(p) - p + 1 also prime, see A234695.

%H Zhi-Wei Sun, <a href="/A235189/b235189.txt">Table of n, a(n) for n = 1..10000</a>

%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014

%e a(10) = 1 since 10 = 3 + 7 with 3, 7 and prime(3) - 3 + 1 = 3 all prime.

%e a(28) = 1 since 28 = 5 + 23 with 5, 23 and prime(5) - 4 = 7 all prime.

%e a(61) = 1 since 61 = 2*7 + 47 with 7, 47 and prime(7) - 6 = 11 all prime.

%e a(98) = 1 since 98 = 31 + 67 with 31, 67 and prime(31) - 30 = 97 all prime.

%t p[n_]:=PrimeQ[Prime[n]-n+1]

%t a[n_]:=Sum[If[p[Prime[k]]&&PrimeQ[n-(1+Mod[n,2])*Prime[k]],1,0],{k,1,PrimePi[(n-1)/2]}]

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

%Y Cf. A000040, A045917, A046927, A234694, A234695, A235187.

%K nonn

%O 1,9

%A _Zhi-Wei Sun_, Jan 04 2014

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Last modified September 29 06:29 EDT 2020. Contains 337425 sequences. (Running on oeis4.)