A257928 Least prime p such that pi(p*n) = pi(q*n)*pi(r*n) for some primes q and r with p > q > r, where pi(x) denotes the number of primes not exceeding x.

13, 7, 13, 67, 19, 79, 47, 193, 107, 41, 229, 179, 383, 281, 173, 1327, 193, 701, 1429, 211, 113, 73, 1093, 83, 1447, 659, 197, 719, 331, 761, 1171, 2269, 467, 509, 863, 113, 643, 577, 563, 379, 607, 1291, 283, 3593, 2549, 881, 1523, 4663, 2657, 3583, 8807, 683, 2251, 863, 8929, 163, 6737, 2459, 4919, 6553

Offset: 1

Author

Zhi-Wei Sun, Jul 13 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. Also, for each positive integer n there are distinct primes p, q and r such that pi(p*n) = pi(q*n) + pi(r*n).

Examples

			a(1) = 13 since 3, 5 and 13 are distinct primes with pi(13*1) = 6 = 2*3 = pi(3*1)*pi(5*1).
a(200) = 105227 since 19, 113 and 105227 are distinct primes with pi(105227*200) = 1332672 = 528*2524 = pi(19*200)*pi(113*200).
		

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..200
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000720, A257364, A257938.

Programs

Mathematica

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

PARI

a(n)={my(i,j,k=3);while(1,for(j=2,k-1,for(i=1,j-1,if(primepi(prime(k)*n) == primepi(prime(i)*n)*primepi(prime(j)*n),break(3));));k++);return(prime(k));} main(size)={return(vector(size,n,a(n)));} /* Anders Hellström, Jul 13 2015 */