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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 (list; graph; refs; listen; history; text; internal format)
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

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..300

Zhi-Wei Sun, Checking the conjecture for n = 1..10 and r = a/b with a,b = 1..100

Zhi-Wei 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

Zhi-Wei Sun, Jul 20 2015

STATUS

approved

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Last modified December 2 11:26 EST 2021. Contains 349440 sequences. (Running on oeis4.)