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

13, 53, 73, 131, 199, 277, 281, 283, 313, 353, 641, 643, 647, 701, 773, 839, 887, 977, 1033, 1103, 1117, 1163, 1187, 1223, 1259, 1409, 1433, 1439, 1487, 1489, 1583, 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

Offset: 1

Author

Zhi-Wei Sun, Sep 23 2015

Keywords

• nonn

Comments

Conjecture: The sequence has infinitely many terms.

See also A262408 and A262443 for related conjectures.

Examples

			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.
		

References

• 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.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..500 from Zhi-Wei Sun)
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

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

Programs

Mathematica

f[n_]:=PrimePi[Prime[n]^2]
T[n_]:=Table[f[k],{k,1,n}]
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}]