

A260197


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


3



5, 277, 29, 17, 43, 103, 53, 31, 1571, 3089, 37, 593, 881, 3023, 277, 9257, 47, 1949, 9137, 311, 17011, 1039, 53, 59, 2153, 15331, 3617, 631, 44867, 61, 17351, 661, 821, 2339, 683, 1201, 34759, 62687, 20327, 59369, 71, 883, 40189, 9187, 1879, 7669, 2767, 3931, 8867, 8081, 79, 12401, 139, 4787, 6367, 277, 2903, 23671, 32839, 3659
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OFFSET

1,1


COMMENTS

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


REFERENCES

ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28  Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.


LINKS



EXAMPLE

a(1) = 5 since pi(5*1) = 3 = prime(2*1) with 2 and 5 both prime.
a(2) = 277 since pi(277*2) = 101 = prime(13*2) with 13 and 277 both prime.
a(10) = 3089 since pi(3089*10) = 3331 = prime(47*10) with 47 and 3089 both prime.


MATHEMATICA

PQ[n_, p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]
Do[k=0; Label[aa]; k=k+1; If[PQ[n, PrimePi[Prime[k]*n]], Goto[bb], Goto[aa]]; Label[bb]; Print[n, " ", Prime[k]]; Continue, {n, 1, 60}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



