A260120 -id:A260120 - 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}]

A261354 Primes p such that prime(p)^2 - 2 = prime(q) for some prime q.

Original entry on oeis.org

31, 191, 541, 809, 1153, 1301, 2221, 3037, 3847, 4049, 4159, 5441, 8243, 10177, 12277, 13681, 14783, 15619, 17903, 19463, 20897, 22697, 24517, 25163, 25847, 25849, 26633, 26647, 27329, 27407, 28051, 32653, 35059, 35747, 36341, 36527, 37369, 37811, 38609, 40949, 42737, 46679, 51061, 51607, 54443, 54679, 56113, 57637, 60887, 61493

Offset: 1

Zhi-Wei Sun, Aug 15 2015

nonn

Conjecture: The sequence has infinitely many terms. In general, for any integers a,b,c with a>0 and gcd(a,b,c)=1, if b^2-4*a*c is not a square, a+b+c is odd, and gcd(b,a+c) is not divisible by 3, then there are infinitely many prime pairs {p,q} such that a*prime(p)^2+b*prime(p)+c = prime(q).

			a(1) = 31 since 31 is a prime, and prime(31)^2-2 = 127^2-2 = 16127 = prime(1877) with 1877 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..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A049002, A062326, A237413, A260120, A261281, A261352, A261361.

PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
f[k_]:=Prime[Prime[k]]^2-2
n=0;Do[If[PQ[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,6200}]

A260140 Least prime p such that pi(pn) = pi(qn)^2 for some prime q, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

2, 5, 19, 3187, 11, 2251, 12149, 19, 239, 23761, 61, 157, 8419, 10973, 1117, 9601, 58741, 37, 53359, 14533, 1063, 934811, 78487, 27647, 1249, 720221, 1616077, 30091, 5501, 131627, 2003, 67, 677, 1313843, 45413, 273943, 127241, 19661, 188317, 811, 33863, 17789, 109073, 602269, 125201, 6424897, 441647, 2512897, 2909, 836471

Offset: 1

Zhi-Wei Sun, Jul 17 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, for any integers a,b,c,n with a > 0 and n > 0, there are two elements x and y of the set {pi(p*n): p is prime} with a*x^2+b*x+c = y.

A supplement to the conjecture: For any integers b,c,n with b > 0 and n > 0, we have b*x+c = y for some elements x and y of the set {pi(p*n): p is prime}. - Zhi-Wei Sun, Aug 02 2015

			a(1) = 2 since pi(2*1) = 1^2 = pi(2*1)^2 with 2 prime.
a(4) = 3187 since pi(3187*4) = 1521 = 39^2 = pi(43*4)^2 with 43 and 3187 both prime.
a(72) = 25135867 since pi(25135867*72) = 89321401 = 9451^2 = pi(1367*72)^2 with 1367 and 25135867 both prime.
a(84) = 106788581 since pi(106788581*84) = 410224516 = 20254^2 = prime(2713*84)^2 with 2713 and 106788581 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..150
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000290, A000720, A257364, A257928, A260120.

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n]
Do[k=0;Label[bb];k=k+1;If[SQ[f[Prime[k]*n]]==False,Goto[bb]];Do[If[Sqrt[f[Prime[k]*n]]==f[Prime[j]*n],Goto[aa]];If[Sqrt[f[Prime[k]*n]]

A260121 Least positive integer k such that prime(kn)^2 - 2 = prime(jn) for some j > 0.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 45, 1, 15, 34, 9, 146, 63, 128, 9, 20, 79, 45, 242, 50, 44, 71, 103, 181, 98, 208, 5, 180, 162, 299, 710, 10, 3, 388, 144, 427, 225, 121, 79, 25, 580, 230, 471, 46, 3, 1040, 11, 224, 305, 56, 1163, 104, 93, 193, 55, 90, 88, 521, 898, 218

Offset: 1

Zhi-Wei Sun, Jul 17 2015

nonn

The conjecture in A260120 implies that a(n) exists for any n > 0, which is stronger than the conjecture in A253257.

			a(5) = 4 since prime(4*5)^2-2 = 71^2-2 = 5039 = prime(135*5).

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..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A062326, A253257, A260080, A260120.

P[n_,p_]:=PrimeQ[p]&&Mod[PrimePi[p],n]==0
Do[k=0;Label[bb];k=k+1; If[P[n,Prime[k*n]^2-2],Goto[aa]];Goto[bb];Label[aa];Print[n, " ", k];Continue,{n,1,60}]

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}]

A261339 Least positive integer k such that both k and k*n belong to the set {m>0: m+1, m^2+1 and m^2+prime(m)^2 are all prime}.

Original entry on oeis.org

1, 1, 47500, 20440, 2, 124560, 17850, 2730, 185550, 1, 518910, 429180, 10, 687480, 81030, 36, 1568340, 2, 1165750, 7410, 10, 6780, 481140, 10, 10, 5430, 240, 2730, 72660, 2080, 18700, 291720, 295080, 52860, 5430, 1, 81030, 56400, 12490, 43590, 124560, 40030, 5170, 278700, 2091850, 131320, 184110, 11206510, 12910, 1245780

Offset: 1

Zhi-Wei Sun, Aug 15 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n with m and n in the set {k>0: k+1, k^2+1 and k^2+prime(k)^2 are all prime}.
For example, 5/8 = 3567600/5708160 with 3567600+1, 3567600^2+1 = 12727769760001, 3567600^2 + prime(3567600)^2 = 3567600^2 + 60098671^2 = 3624578025726241, 5708160+1, 5708160^2+1 = 32583090585601,  and 5708160^2 + prime(5708160)^2 = 5708160^2 + 99018553^2 = 9837256928799409 all prime.
The conjecture implies that there are infinitely many primes p with (p-1)^2+1 and (p-1)^2+prime(p-1)^2 both prime.
We also guess that any positive rational number can be written as m/n, where m and n are positive integers with m^2+prime(m)^2, m^2+prime(n)^2, n^2+prime(m)^2 and n^2+prime(n)^2 all prime.

			a(3) = 47500 since 47501, 47500^2 + 1 = 2256250001, 47500^2 + prime(47500)^2 = 47500^2 + 578827^2 = 337296945929, 47500*3 + 1 = 142501, (47500*3)^2 + 1 = 20306250001, and (47500*3)^2 + prime(47500*3)^2 = 142500^2 + 1907023^2 = 3657042972529 are all 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..1000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..60
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A005574, A236193, A259492, A259540, A260120.

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

A260219 Least prime p such that pi(pn)^2 + 1 = prime(qn) for some prime q.

Original entry on oeis.org

3, 47, 229, 379, 11, 2687, 1181, 24547, 5, 509, 5, 16619, 877, 22543, 3, 9067, 60337, 10667, 997, 24061, 18329, 56099, 1787, 58757, 108883, 416881, 157141, 11003, 14939, 113167, 101957, 77969, 613, 27947, 52153, 158551, 6197, 36607, 25237, 27179, 330689, 203617, 77419, 708269, 87649, 340381, 267601, 67153, 123377, 21617

Offset: 1

Zhi-Wei Sun, Jul 19 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, if a,b,c are integers with a > 0 and gcd(a,b,c) = 1, and a+b or c is odd, and b^2 - 4*a*c is not a square, then there are primes p and q such that a*pi(p*n)^2 + b*pi(p*n) + c = prime(q*n).

			a(1) = 3 since pi(3*1)^2 + 1 = 5 = prime(3*1) with 3 prime.
a(8) = 24547 since pi(24547*8)^2 + 1 = 17686^2 + 1 = 312794597 = prime(2113417*8) with 24547 and 2113417 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..72
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000720, A002496, A259731, A260120, A260197.

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

first(m)={my(v=vector(m),p,q,n);for(n=1,m,p=0;while(1,p++;q=1;while(primepi(prime(p)*n)^2 +1 >= prime(prime(q)*n), if(primepi(prime(p)*n)^2 +1 == prime(prime(q)*n),v[n]=prime(p);break(2),q++;))));v;} /* Anders Hellström, Jul 19 2015 */

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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A261354 Primes p such that prime(p)^2 - 2 = prime(q) for some prime q.

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A260140 Least prime p such that pi(p*n) = pi(q*n)^2 for some prime q, where pi(x) denotes the number of primes not exceeding x.

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A260121 Least positive integer k such that prime(k*n)^2 - 2 = prime(j*n) for some j > 0.

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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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A261339 Least positive integer k such that both k and k*n belong to the set {m>0: m+1, m^2+1 and m^2+prime(m)^2 are all prime}.

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A260219 Least prime p such that pi(p*n)^2 + 1 = prime(q*n) for some prime q.

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PARI

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

A260140 Least prime p such that pi(pn) = pi(qn)^2 for some prime q, where pi(x) denotes the number of primes not exceeding x.

A260121 Least positive integer k such that prime(kn)^2 - 2 = prime(jn) for some j > 0.

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

A260219 Least prime p such that pi(pn)^2 + 1 = prime(qn) for some prime q.