

A260232


Least prime p such that pi(p*n) = n*pi(q*n) for some prime q.


3



2, 5, 13, 67, 23, 19, 433, 443, 107, 41, 61, 251, 239, 1987, 541, 491, 1093, 499, 421, 179, 2137, 1297, 1097, 101, 103, 2411, 1283, 1847, 379, 4993, 8329, 5563, 4297, 5639, 9587, 1867, 5113, 6691, 3691, 1193, 4663, 2971, 27733, 7121, 593, 2273, 607, 6047, 4217, 2609
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OFFSET

1,1


COMMENTS

Conjecture: For any positive integer n, each rational number r > 0 can be written as pi(p*n)/pi(q*n) with p and q both prime.
For example, 4/7 = 416/728 = pi(479*6)/pi(919*6) with 479 and 919 both prime.
The conjecture holds trivially for n = 1 since pi(prime(m)*1) = m for all m = 1,2,3,.... Also, the conjecture implies that a(n) exists for any n > 0.


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

ZhiWei Sun, Table of n, a(n) for n = 1..300
ZhiWei Sun, Checking the conjecture for n = 1..10 and r = a/b with a,b = 1..100
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


EXAMPLE

a(4) = 67 since pi(67*4) = 56 = 4*14 = 4*pi(11*4) with 11 and 67 both prime.


MATHEMATICA

f[n_]:=PrimePi[n]; Do[k=0; Label[bb]; k=k+1; If[Mod[f[Prime[k]*n], n]>0, Goto[bb]]; Do[If[f[Prime[k]n]==n*f[Prime[j]*n], Goto[aa]]; If[f[Prime[k]n]<n*f[Prime[j]*n], Goto[bb]]; Continue, {j, 1, k}]; Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 50}]


CROSSREFS

Cf. A000040, A000720, A257364, A257928, A260140, A260252.
Sequence in context: A306827 A028856 A013497 * A262366 A128029 A215167
Adjacent sequences: A260229 A260230 A260231 * A260233 A260234 A260235


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jul 20 2015


STATUS

approved



