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

1,1

Comments

Conjecture: The sequence has infinitely many terms.

See also A262408 and A262443 for related conjectures.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A207977 A056255 A141885 * A165352 A262287 A307749

Adjacent sequences: A262444 A262445 A262446 * A262448 A262449 A262450

Keyword

nonn

Author

Zhi-Wei Sun, Sep 23 2015

Status

approved