A257364 Least prime p such that pi(p*n)^2 = pi(q*n)^2 + pi(r*n)^2 for some primes q and r, where pi(x) denotes the number of primes not exceeding x.

11, 59, 47, 211, 23, 233, 181, 257, 109, 109, 13, 311, 929, 47, 389, 757, 1747, 13, 67, 2389, 1087, 569, 311, 853, 103, 5569, 1399, 3203, 10891, 3673, 3793, 1873, 4357, 41, 2297, 131, 3253, 6737, 2621, 5113, 2879, 953, 6379, 3539, 12343, 4337, 6067, 11939, 43441, 5179

Offset: 1

Author

Zhi-Wei Sun, Jul 11 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. In other words, for each fixed positive integer n the sequence pi(p*n) with p prime contains a Pythagorean triple.

This is stronger than the conjecture in A255679.

Examples

			a(1) = 11 since 5, 7 and 11 are primes with pi(5*1)^2 + pi(7*1)^2 = 3^2 + 4^2 = 5^2 = pi(11*1)^2.
a(45) = 12343 since 4337, 11311 and 12343 are primes with pi(4337*45)^2 + pi(11311*45)^2 = 17590^2 + 42216^2 = 45734^2 = pi(12343*45)^2.
a(49) = 43441 since 15427, 39839 and 43441 are primes with pi(15427*49)^2 + pi(39839*49)^2 = 60685^2 + 145644^2 = 157781^2 = pi(43441*49)^2.
		

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

• Zhi-Wei Sun, Table of n, a(n) for n = 1..100
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000720, A255679, A257928.

Programs

Mathematica

f[n_]:=PrimePi[n]
Do[k=0;Label[bb];k=k+1;Do[Do[If[f[Prime[k]*n]^2==f[Prime[i]*n]^2+f[Prime[j]*n]^2,Goto[aa]];If[f[Prime[k]*n]^2