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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.
2
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.
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