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A263100 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2) + pi(m^2/2), where pi(x) denotes the number of primes not exceeding x.

%I A263100 #12 Oct 09 2015 09:42:07
%S A263100 1,2,1,3,2,3,3,2,4,2,6,2,5,2,5,4,4,4,4,5,3,5,5,4,4,6,6,1,7,4,6,4,4,7,
%T A263100 6,4,5,5,5,6,6,4,6,3,7,6,5,6,6,6,5,5,6,4,7,8,4,3,10,2,6,6,6,6,7,5,5,9,
%U A263100 3,6,8,6,7,5,5,6,7,7,8,3,9,3,10,2,7,9,7,2,7,8,5,8,4,6,9,5,7,6,5,7
%N A263100 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2) + pi(m^2/2), where pi(x) denotes the number of primes not exceeding x.
%C A263100 Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 28.
%C A263100 (ii) Any integer n > 0 can be written as pi(k^2) + pi((m^2+1)/2) with k and m positive integers.
%C A263100 (iii) Each n = 1,2,3,... can be written as pi(k^2/2) + pi((m^2+1)/2) with k and m positive integers.
%C A263100 See also A262995, A262999, A263001 and A263020 for similar conjectures.
%H A263100 Zhi-Wei Sun, <a href="/A263100/b263100.txt">Table of n, a(n) for n = 1..10000</a>
%e A263100 a(1) = 1 since 1 = 0 + 1 = pi(1^2) + pi(2^2/2).
%e A263100 a(3) = 1 since 3 = 2 + 1 = pi(2^2) + pi(2^2/2).
%e A263100 a(28) = 1 since 28 = 11 + 17 = pi(6^2) + pi(11^2/2).
%t A263100 s[n_]:=s[n]=PrimePi[n^2]
%t A263100 t[n_]:=t[n]=PrimePi[n^2/2]
%t A263100 Do[r=0; Do[If[s[k]>n, Goto[bb]]; Do[If[t[j]>n-s[k], Goto[aa]]; If[t[j]==n-s[k], r=r+1]; Continue, {j, 1, n-s[k]+1}]; Label[aa]; Continue, {k, 1, n}];
%t A263100 Label[bb]; Print[n, " ", r]; Continue, {n,1,100}]
%Y A263100 Cf. A000290, A000720, A038107, A262995, A262999, A263001, A263020.
%K A263100 nonn
%O A263100 1,2
%A A263100 _Zhi-Wei Sun_, Oct 09 2015