cp's OEIS Frontend

A263020 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k*(k+1)/2) + pi(m*(3*m-1)/2), where pi(x) denotes the number of primes not exceeding x.

%I A263020 #48 Oct 19 2015 05:12:19
%S A263020 0,1,2,1,2,2,2,3,3,1,5,2,2,5,2,3,4,2,6,1,5,3,3,5,2,4,5,2,4,5,1,6,5,2,
%T A263020 6,4,3,5,4,5,3,6,4,4,4,5,4,5,4,5,6,2,3,7,5,3,6,5,2,3,8,5,3,5,5,6,5,1,
%U A263020 8,8,2,4,6,6,3,5,8,4,4,5,3,9,2,6,8,3,3,6,4,7,3,6,6,5,5,5,3,7,6,6
%N A263020 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k*(k+1)/2) + pi(m*(3*m-1)/2), where pi(x) denotes the number of primes not exceeding x.
%C A263020 Conjecture: a(n) > 0 for all n > 1.
%C A263020 We have verified this for n up to 3*10^5. It seems that a(n) = 1 only for n = 2, 4, 10, 20, 31, 68, 147, 252, 580, 600, 772, 1326, 1381, 2779, 3136, 3422, 3729, 7151, 9518, 13481, 18070, 18673, 36965, 48181, 69250, 91130, 93580, 99868.
%C A263020 Note that n*(n+1)/2 (n = 0,1,2,...) are the triangular numbers while n*(3n-1)/2 (n = 0,1,2,...) are the pentagonal numbers.
%H A263020 Zhi-Wei Sun, <a href="/A263020/b263020.txt">Table of n, a(n) for n = 1..10000</a>
%H A263020 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;313507b8.1510">Conjectures on the prime-counting function</a>, a message to Number Theory Mailing List, Oct. 19, 2015.
%e A263020 a(2) = 1 since 2 = 2 + 0 = pi(2*3/2) + pi(1*(3*1-1)/2).
%e A263020 a(4) = 1 since 4 = 4 + 0 = pi(4*5/2) + pi(1*(3*1-1)/2).
%e A263020 a(10) = 1 since 10 = 2 + 8 = pi(2*3/2) + pi(4*(3*4-1)/2).
%e A263020 a(20) = 1 since 20 = 9 + 11 = pi(7*8/2) + pi(5*(3*5-1)/2).
%e A263020 a(31) = 1 since 31 = 16 + 15 = pi(10*11/2) + pi(6*(3*6-1)/2).
%e A263020 a(68) = 1 since 68 = 2 + 66 = pi(2*3/2) + pi(15*(3*15-1)/2).
%e A263020 a(147) = 1 since 147 = pi(31*32/2) + pi(13*(3*13-1)/2).
%e A263020 a(252) = 1 since 252 = pi(29*30/2) + pi(26*(3*26-1)/2).
%e A263020 a(580) = 1 since 580 = pi(5*6/2) + pi(53*(3*53-1)/2).
%e A263020 a(600) = 1 since 600 = pi(42*43/2) + pi(46*(3*46-1)/2).
%e A263020 a(772) = 1 since 772 = pi(107*108/2) + pi(6*(3*6-1)/2).
%e A263020 a(1326) = 1 since 1326 = pi(139*140/2) + pi(22*(3*22-1)/2).
%e A263020 a(1381) = 1 since 1381 = pi(145*146/2) + pi(18*(3*18-1)/2).
%e A263020 a(2779) = 1 since 2779 = pi(212*213/2) + pi(33*(3*33-1)/2).
%e A263020 a(3136) = 1 since 3136 = pi(147*148/2) + pi(102*(3*102-1)/2).
%e A263020 a(3422) = 1 since 3422 = pi(151*152/2) + pi(109*(3*109-1)/2).
%e A263020 a(3729) = 1 since 3729 = pi(29*30/2) + pi(151*(3*151-1)/2).
%e A263020 a(7151) = 1 since 7151 = pi(100*101/2) + pi(208*(3*208-1)/2).
%e A263020 a(9518) = 1 since 9518 = pi(82*83/2) + pi(250*(3*250-1)/2).
%e A263020 a(13481) = 1 since 13481 = pi(539*540/2) + pi(6*(3*6-1)/2).
%e A263020 a(18070) = 1 since 18070 = pi(632*633/2) + pi(17*(3*17-1)/2).
%e A263020 a(18673) = 1 since 18673 = 14493 + 4180 = pi(561*562/2) + pi(163*(3*163-1)/2).
%e A263020 a(36965) = 1 since 36965 = 3780 + 33185 = pi(266*267/2) + pi(511*(3*511-1)/2).
%e A263020 a(48181) = 1 since 48181 = 30755 + 17426 = pi(848*849/2) + pi(359*(3*359-1)/2).
%e A263020 a(69250) = 1 since 69250 = 20669 + 48581 = pi(682*683/2) + pi(629*(3*629-1)/2).
%e A263020 a(91130) = 1 since 91130 = 81433 + 9697 = pi(1442*1443/2) + pi(260*(3*260-1)/2).
%e A263020 a(93580) = 1 since 93580 = 91865 + 1715 = pi(1539*1540/2) + pi(99*(3*99-1)/2).
%e A263020 a(99868) = 1 since 99868 = 66079 + 33789 = pi(1287*1288/2) + pi(516*(3*516-1)/2).
%t A263020 s[n_]:=s[n]=PrimePi[n(3n-1)/2]
%t A263020 t[n_]:=t[n]=PrimePi[n(n+1)/2]
%t A263020 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 A263020 Cf. A000217, A000326, A000720, A111208, A262403, A262995, A262999, A263001, A263107.
%K A263020 nonn
%O A263020 1,3
%A A263020 _Zhi-Wei Sun_, Oct 07 2015