A260888 Least prime p such that 2 + 3*pi(p*n) = 4*pi(q*n) for some prime q, where pi(x) denotes the number of primes not exceeding x.

3, 2, 41, 211, 23, 83, 43, 23, 7, 3, 601, 109, 23, 251, 31, 251, 7, 41, 149, 157, 293, 3, 103, 41, 2083, 233, 7, 647, 1877, 7, 1117, 599, 7, 937, 487, 7, 251, 149, 7, 439, 83, 3, 7, 43, 643, 7, 157, 157, 1291, 7

Offset: 1

Author

Zhi-Wei Sun, Aug 02 2015

Keywords

• nonn

Comments

Conjecture: Let a and b be relatively prime positive integers, and let c be any integer. For any positive integer n, there are primes p and q such that a*pi(p*n) - b*pi(q*n) = c.

In the case c = 0, this reduces to the conjecture in A260232.

For example, for a = 20, b = 19, c = 18 and n = 28, we have 20*pi(4549*28)-19*pi(4813*28) = 20*11931-19*12558 = 18 with 4549 and 4813 both prime.

Examples

			a(5) = 23 since 2+3*pi(23*5) = 2+3*30 = 92 = 4*23 = 4*pi(17*5) with 23 and 17 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..600
• Zhi-Wei Sun, Checking the conjecture for a,b,c = 1..20 and n = 1..30
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2015.

Crossrefs

Cf. A000040, A000720, A260140, A260232, A260886.

Programs

Mathematica

f[k_,n_]:=PrimePi[Prime[k]*n]
Do[k=0;Label[bb];k=k+1;If[Mod[3*f[k,n]+2,4]>0,Goto[bb]];Do[If[(3*f[k,n]+2)/4==f[j,n],Goto[aa]];If[(3*f[k,n]+2)/4