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%I #74 Jul 02 2018 22:05:20
%S 0,0,1,2,2,3,2,3,4,2,3,2,2,3,2,4,3,2,3,3,3,1,3,3,1,4,4,2,3,4,4,4,4,5,
%T 3,4,4,1,4,4,3,5,4,3,3,4,6,3,5,5,3,3,3,2,4,5,4,5,4,2,3,4,4,5,5,7,4,5,
%U 2,6,4,5,7,3,5,6,2,4,3,2
%N Number of ordered ways to write n = k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(m)) + 2 are both prime.
%C Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 22, 25, 38, 101, 273.
%C (ii) Each n = 2, 3, ... can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1 and prime(prime(m)) + 2 are all prime.
%C (iii) Any integer n > 5 can be written as k + m with k > 0 and m > 0 such that phi(k) - 1, phi(k) + 1 and prime(prime(m)) + 2 are all prime, where phi(.) is Euler's totient function.
%C (iv) If n > 2 is neither 10 nor 31, then n can be written as k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(prime(m))) + 2 are both prime.
%C (v) If n > 1 is not equal to 133, then n can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1 and prime(prime(prime(m))) + 2 are all prime.
%C Clearly, each part of the conjecture implies the twin prime conjecture.
%C We have verified part (i) for n up to 10^9. See the comments in A237348 for an extension of this part.
%H Zhi-Wei Sun, <a href="/A218829/b218829.txt">Table of n, a(n) for n = 1..10000</a>
%H Andrei-Lucian Dragoi, <a href="https://doi.org/10.9734/JAMCS/2017/36895">The "Vertical" Generalization of the Binary Goldbach's Conjecture as Applied on "Iterative" Primes with (Recursive) Prime Indexes (i-primeths)</a>, Journal of Advances in Mathematics and Computer Science (2017), Vol. 25, No. 2, pp. 1-32.
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;b634e7c8.1401">Unification of Goldbach's conjecture and the twin prime conjecture</a>, a message to Number Theory List, Jan. 29, 2014.
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;b81b9aa9.1402">Super Twin Prime Conjecture</a>, a message to Number Theory List, Feb. 6, 2014.
%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(3) = 1 since 3 = 2 + 1 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(1)) + 2 = prime(2) + 2 = 5 both prime.
%e a(22) = 1 since 22 = 20 + 2 with prime(20) + 2 = 71 + 2 = 73 and prime(prime(2)) + 2 = prime(3) + 2 = 5 + 2 = 7 both prime.
%e a(25) = 1 since 25 = 2 + 23 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(23)) + 2 = prime(83) + 2 = 431 + 2 = 433 both prime.
%e a(38) = 1 since 38 = 35 + 3 with prime(35) + 2 = 149 + 2 = 151 and prime(prime(3)) + 2 = prime(5) + 2 = 11 + 2 = 13 both prime.
%e a(101) = 1 since 101 = 98 + 3 with prime(98) + 2 = 521 + 2 = 523 and prime(prime(3)) + 2 = prime(5) + 2 = 11 + 2 = 13 both prime.
%e a(273) = 1 since 273 = 2 + 271 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(271)) + 2 = prime(1741) + 2 = 14867 + 2 = 14869 both prime.
%t pq[n_]:=PrimeQ[Prime[n]+2]
%t PQ[n_]:=PrimeQ[Prime[Prime[n]]+2]
%t a[n_]:=Sum[If[pq[k]&&PQ[n-k],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000010, A000040, A001359, A002822, A006512, A072281, A182662, A199920, A236531, A236566, A236831, A236968, A237127, A237130, A237168, A237253, A237259, A237260, A237348.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Feb 05 2014
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