cp's OEIS Frontend

A262408 Positive integers m such that pi(m^2) = pi(j^2) + pi(k^2) for no 0 < j <= k < m.

%I A262408 #51 Sep 24 2015 08:04:05
%S A262408 1,2,5,10,21,23,46,103,105,193,222,232,285,309,345,392,404,476,587,
%T A262408 670,779,912,1086,1162,1249,2508,2592,2852,2964,3362,3673,3895,4218,
%U A262408 4732,5452,6417,7667,7759,8430,8796,9606,11096,11953,12014,12125,13956,14474,15018,17854,18861,18879,19307,22843,28106,29423,31576,37182
%N A262408 Positive integers m such that pi(m^2) = pi(j^2) + pi(k^2) for no 0 < j <= k < m.
%C A262408 Conjecture: (i) There are infinitely many positive integers m such that pi(m^2) = pi(x^2) + pi(y^2) for some 0 < x <= y < m. Also, the current sequence has infinitely many terms.
%C A262408 (ii) For every n = 4,5,... the equation pi(x^n) + pi(y^n) = pi(z^n) has no integral solution with 0 < x <= y < z.
%C A262408 It is interesting to compare this conjecture with Fermat's Last Theorem. See also A262409 for the equation pi(x^3) + pi(y^3) = pi(z^3).
%D A262408 Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H A262408 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e A262408 a(3) = 5 since pi(5^2) = 9 cannot be written as pi(j^2) + pi(k^2) with 0 < j <= k < 5. Note that pi(1^2) = 0, pi(2^2) = 2, pi(3^2) = 4 and pi(4^2) = 6 are all even.
%t A262408 f[n_]:=PrimePi[n^2]
%t A262408 T[n_]:=Table[f[k],{k,1,n}]
%t A262408 n=0;Do[Do[If[MemberQ[T[m-1],f[m]-f[k]],Goto[aa]],{k,1,m-1}];n=n+1;Print[n," ",m];Label[aa];Continue,{m,1,32000}]
%Y A262408 Cf. A000720, A038107, A262403, A262409.
%K A262408 nonn
%O A262408 1,2
%A A262408 _Zhi-Wei Sun_, Sep 21 2015