OFFSET
1,1
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).
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.
EXAMPLE
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).
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]<f[Prime[i]*n]*f[Prime[j]*n], Goto[cc]]; Continue, {i, 1, j-1}]; Label[cc]; Continue, {j, 1, k-1}]; Goto[bb];
Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 60}]
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 13 2015
STATUS
approved