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%I #11 Oct 08 2015 04:25:56
%S 1,0,2,1,3,1,3,2,3,3,3,4,3,4,2,5,4,2,7,2,4,5,2,7,2,5,4,4,5,3,5,6,4,5,
%T 6,3,6,6,2,9,3,5,5,5,6,5,6,5,4,7,4,7,4,5,6,7,3,5,6,7,4,7,7,5,3,9,5,7,
%U 3,8,7,5,4,8,6,6,3,10,7,3,3,11,5,7,4,8,5,4,7,7,5,8,3,8,7,4,5,9,6,9
%N Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k*(k+1)) + pi(m*(m+1)/2), where pi(x) denotes the number of primes not exceeding x.
%C Conjecture: a(n) > 0 for all n > 2, and a(n) = 1 only for n = 1, 4, 6.
%C We have verified this for n up to 10^5.
%C See also A262995, A262999 and A263020 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A263001/b263001.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 1 since 1 = pi(1*2) + pi(1*2/2).
%e a(4) = 1 since 4 = pi(1*2) + pi(3*4/2).
%e a(6) = 1 since 6 = pi(2*3) + pi(3*4/2).
%t s[n_]:=s[n]=PrimePi[n(n+1)]
%t t[n_]:=t[n]=PrimePi[n(n+1)/2]
%t 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}];Label[bb];Print[n," ",r];Continue,{n,1,100}]
%Y Cf. A000217, A000720, A002378, A111208, A262995, A262999, A263020.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Oct 07 2015
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