A260252 -id:A260252 - cp's OEIS frontend

A260886 Least prime p such that 3 + 4prime(pn) = 5prime(qn) 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

Zhi-Wei Sun, Aug 02 2015

nonn

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.

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

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.

Cf. A000040, A260120, A260252, A260882.

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}]

A260232 Least prime p such that pi(pn) = npi(q*n) for some prime q.

Original entry on oeis.org

2, 5, 13, 67, 23, 19, 433, 443, 107, 41, 61, 251, 239, 1987, 541, 491, 1093, 499, 421, 179, 2137, 1297, 1097, 101, 103, 2411, 1283, 1847, 379, 4993, 8329, 5563, 4297, 5639, 9587, 1867, 5113, 6691, 3691, 1193, 4663, 2971, 27733, 7121, 593, 2273, 607, 6047, 4217, 2609

Offset: 1

Zhi-Wei Sun, Jul 20 2015

nonn

Conjecture: For any positive integer n, each rational number r > 0 can be written as pi(p*n)/pi(q*n) with p and q both prime.
For example, 4/7 = 416/728 = pi(479*6)/pi(919*6) with 479 and 919 both prime.
The conjecture holds trivially for n = 1 since pi(prime(m)*1) = m for all m = 1,2,3,.... Also, the conjecture implies that a(n) exists for any n > 0.

			a(4) = 67 since pi(67*4) = 56 = 4*14 = 4*pi(11*4) with 11 and 67 both prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..300
Zhi-Wei Sun, Checking the conjecture for n = 1..10 and r = a/b with a,b = 1..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000720, A257364, A257928, A260140, A260252.

f[n_]:=PrimePi[n]; Do[k=0;Label[bb];k=k+1;If[Mod[f[Prime[k]*n],n]>0,Goto[bb]];Do[If[f[Prime[k]n]==n*f[Prime[j]*n],Goto[aa]];If[f[Prime[k]n]

A260882 Least prime p such that 2prime(pn)+1 = prime(q*n) for some prime q.

Original entry on oeis.org

3, 47, 3, 13, 797, 89, 2269, 733, 7877, 53, 14683, 16267, 17167, 59951, 10067, 761, 94463, 12437, 124561, 71881, 52009, 6791, 10061, 47287, 10789, 19009, 4813, 23173, 27427, 18701, 23011, 44917, 17, 70937, 883, 727, 99079, 10531, 18749, 126541, 18121, 34807, 29873, 159473, 853, 165317, 80627, 159721, 8263, 411707

Offset: 1

Zhi-Wei Sun, Aug 02 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, if a > 1 and b are integers with a+b odd and gcd(a,b)=1, then for any positive integer n there are primes p and q such that a*prime(p*n)+b = prime(q*n).

This is a supplement to the conjecture in A260120. It implies that there are infinitely many Sophie Germain primes.

			a(2) = 47 since 2*prime(47*2)+1 = 2*491+1 = 983 = prime(83*2) with 47 and 83 both prime.
a(199) = 2784167 since 2*prime(2784167*199)+1 = 2*12290086499+1 = 24580172999 = prime(5399231*199) with 2784167 and 5399231 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2015.

Cf. A000040, A005384, A260120, A260252.

f[n_]:=Prime[n]
PQ[p_,n_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]
Do[k=0;Label[bb];k=k+1;If[PQ[2*f[n*f[k]]+1,n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", f[k]];Continue,{n,1,50}]

cp's OEIS Frontend

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

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A260232 Least prime p such that pi(pn) = npi(q*n) for some prime q.

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A260882 Least prime p such that 2prime(pn)+1 = prime(q*n) for some prime q.

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cp's OEIS Frontend

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

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A260232 Least prime p such that pi(p*n) = n*pi(q*n) for some prime q.

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A260882 Least prime p such that 2*prime(p*n)+1 = prime(q*n) for some prime q.

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

A260232 Least prime p such that pi(pn) = npi(q*n) for some prime q.

A260882 Least prime p such that 2prime(pn)+1 = prime(q*n) for some prime q.