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%I #14 Aug 05 2019 09:26:10
%S 0,0,0,1,1,1,1,2,1,1,1,1,2,3,1,1,2,2,2,3,2,2,2,3,3,2,2,1,2,4,3,3,4,2,
%T 1,2,3,4,3,2,2,4,4,4,3,2,3,6,4,3,5,2,2,5,3,4,4,2,3,5,5,5,4,2,3,6,4,4,
%U 4,3,4,6,6,6,5,2,3,5,5,7,6,4,4,5,6,6,3,3,7,7,5,4,5,4,5,6,2,6,6,4
%N Number of ways to write n = p + q - pi(q), where p and q are odd primes not exceeding n, and pi(q) is the number of primes not exceeding q.
%C Note that this sequence is different from A232443.
%C Conjecture: a(n) > 0 for all n > 3. Also, a(n) = 1 only for n = 4, 5, 6, 7, 9, 10, 11, 12, 15, 16, 28, 35.
%H Zhi-Wei Sun, <a href="/A232463/b232463.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588 [math.NT], 2012-2017.
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n+ bn modulo m</a>, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
%e a(10) = 1 since 10 = 7 + 7 - pi(7), and 7 is an odd prime not exceeding 10.
%e a(11) = 1 since 11 = 5 + 11 - pi(11), and 5 and 11 are odd primes not exceeding 11.
%e a(15) = 1 since 15 = 13 + 5 - pi(5), and 13 and 5 are odd primes not exceeding 15.
%e a(28) = 1 since 28 = 17 + 19 - pi(19), and 17 and 19 are odd primes not exceeding 28.
%e a(35) = 1 since 35 = 29 + 11 - pi(11), and 29 and 11 are odd primes not exceeding 35.
%t PQ[n_]:=n>2&&PrimeQ[n]
%t a[n_]:=Sum[If[PQ[n-Prime[k]+k],1,0],{k,2,PrimePi[n]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000720, A002375, A232398, A232443.
%K nonn
%O 1,8
%A _Zhi-Wei Sun_, Nov 24 2013
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