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A260886 Least prime p such that 3 + 4*prime(p*n) = 5*prime(q*n) for some prime q. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 6 14:47 EST 2019. Contains 329806 sequences. (Running on oeis4.)