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

1,1

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.

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.

Example

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.

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<f[j, n], Goto[bb]]; Continue, {j, 1, k}]; Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 50}]

Crossrefs

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

Sequence in context: A009084 A096057 A179984 * A065085 A321169 A093398

Adjacent sequences: A260885 A260886 A260887 * A260889 A260890 A260891

Keyword

nonn

Author

Zhi-Wei Sun, Aug 02 2015

Status

approved