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A262447 Primes p such that pi(p^2) = pi(q^2) + pi(r^2) for some distinct primes q and r. 7

%I #11 May 31 2018 08:17:06

%S 13,53,73,131,199,277,281,283,313,353,641,643,647,701,773,839,887,977,

%T 1033,1103,1117,1163,1187,1223,1259,1409,1433,1439,1487,1489,1583,

%U 1721,1913,1931,2239,2243,2269,2309,2371,2441,2473,2477,2621,2683,2707,2797,2843,2851,2953,3049,3137,3257,3307,3499,3511,3613,3659,3769,3779,3911

%N Primes p such that pi(p^2) = pi(q^2) + pi(r^2) for some distinct primes q and r.

%C Conjecture: The sequence has infinitely many terms.

%C See also A262408 and A262443 for related conjectures.

%D 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 Chai Wah Wu, <a href="/A262447/b262447.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..500 from Zhi-Wei Sun)

%H 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 a(1) = 13 since pi(13^2) = pi(169) = 39 = 9 + 30 = pi(5^2) + pi(11^2) with 13, 5 and 11 distinct primes.

%t f[n_]:=PrimePi[Prime[n]^2]

%t T[n_]:=Table[f[k],{k,1,n}]

%t n=0;Do[Do[If[2*f[k]>=f[m],Goto[aa]];If[MemberQ[T[m-1],f[m]-f[k]],n=n+1;Print[n," ",Prime[m]];Goto[aa]];Continue,{k,1,m-1}];Label[aa];Continue,{m,1,541}]

%Y Cf. A000040, A000290, A000720, A262408, A262443.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Sep 23 2015

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