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, 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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 02 2015
STATUS
approved