A260886 Least prime p such that 3 + 4*prime(p*n) = 5*prime(q*n) for some prime q.

2, 157, 199, 3539, 1973, 9241, 14629, 167, 48281, 2207, 313, 30631, 35993, 33863, 23, 23, 7963, 17077, 11069, 6043, 4931, 3697, 2339, 14153, 35311, 63149, 111143, 491, 247193, 464237, 2293, 12101, 727, 61403, 243437, 40289, 4337, 241, 2719, 13933, 21817, 6803, 52813, 451279, 166409, 45631, 109891, 490969, 153563, 9127

Offset

1,1

Comments

Conjecture: Let a,b,c be pairwise relatively prime positive integers with a+b+c even and a not equal to b. Then, for any positive integer n, there are primes p and q such that a*prime(p*n) - b*prime(q*n) = c.

This includes the conjectures in A260252 and A260882 as special cases.

For example, for a = 7, b = 17, c = 20 and n = 30, we have 7*prime(4695851*30) - 17*prime(2020243*30) = 7*2922043519 - 17*1203194389 = 20 with 4695851 and 2020243 both prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..200

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(2) = 157 since 3 + 4*prime(157*2) = 3 + 4*2083 = 8335 = 5*prime(131*2) with 157 and 131 both prime.

Mathematica

f[n_]:=Prime[n]
PQ[p_, n_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]
Do[k=0; Label[bb]; k=k+1; If[PQ[(4*f[n*f[k]]+3)/5, n], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", f[k]]; Continue, {n, 1, 50}]

Crossrefs

Cf. A000040, A260120, A260252, A260882.

Sequence in context: A151614 A103042 A284458 * A142006 A233575 A233263

Adjacent sequences: A260883 A260884 A260885 * A260887 A260888 A260889

Keyword

nonn

Author

Zhi-Wei Sun, Aug 02 2015

Status

approved