A258803 -id:A258803 - cp's OEIS frontend

A259487 Least positive integer m with prime(m)+2 and prime(prime(m))+2 both prime such that prime(mn)+2 and prime(prime(mn))+2 are both prime.

2, 1860, 408, 25011, 51312, 37977, 695, 4071, 10970, 3621, 17671, 12005, 1230, 19494, 542, 577, 408, 2476, 584, 542, 469, 34229, 343, 24078, 3011, 25749, 20706, 24198, 2478, 3926, 1030, 1030, 13857, 3621, 343, 13380, 2476, 4922, 2476, 296, 19176, 29175, 34737, 13, 625, 2956, 408, 572, 7, 469, 15604, 9699, 26515, 2167, 5302, 9773, 54254, 1410, 4524, 4351

Offset: 1

Zhi-Wei Sun, Jun 28 2015

nonn

Conjecture: Any positive rational number r can be written as m/n with m and n terms of A259488.
This implies that there are infinitely many primes p with p+2 and prime(p)+2 both prime.
I have verified the conjecture for all those r = a/b with a,b = 1,...,400. - Zhi-Wei Sun, Jun 29 2015

			a(1) = 2 since prime(2)+2 = 3+2 = 5 and prime(prime(2))+2 = prime(3)+2 = 7 are both prime, but prime(1)+2 = 4 is composite.
a(49) = 7 since prime(7)+2 = 17+2 = 19, prime(prime(7))+2 = prime(17)+2 = 59+2 = 61, prime(49*7)+2 = 2309+2 = 2311 and prime(prime(49*7))+2 = prime(2309)+2 = 20441+2 = 20443 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..5000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..400
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A001359, A006512, A236458, A258803, A258836, A259488, A259492.

PQ[k_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]]+2]
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[n*k], Goto[aa], Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]

A259540 Least positive integer k such that k and k*n are terms of A259539.

Original entry on oeis.org

60, 326940, 728700, 115020, 375258, 70920, 33150, 297990, 2340, 72870, 858, 1416210, 284130, 78978, 91368, 9438, 5547000, 767760, 1182918, 30468, 485208, 60, 7908810, 916188, 21522, 823968, 87720, 390210, 3252, 72870, 7878, 1823010, 1179990, 98010, 3462, 7878, 280590, 6870, 60, 434460

Offset: 1

Zhi-Wei Sun, Jun 30 2015

nonn

Conjecture: Any positive rational number r can be written as m/n with m and n terms of A259539.

For example, 4/5 = 11673840/14592300 with 11673840 and 14592300 terms of A259539.

			a(22) = 60 since 60 and 60*22 = 1320 are terms of A259539. In fact, 60-1 = 59, 60+1 = 61, prime(60)+2 = 283, 1320-1 = 1319, 1320+1 = 1321 and prime(1320)+2 = 10861 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..2000
Zhi-Wei Sun, Checking the conjecture for 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, A014574, A258803, A258836, A259487, A259492, A259539.

PQ[k_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]+1]+2]
QQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[Prime[n]+2]
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&QQ[n*(Prime[k]+1)], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", Prime[k]+1];Continue,{n, 1, 40}]

A258836 Least practical number q with q-1 and q+1 twin prime such that n = q'/q for some practical number q' with q'-1 and q'+1 twin prime.

Original entry on oeis.org

4, 6, 4, 18, 6, 12, 6, 30, 12, 6, 18, 6, 150, 30, 4, 12, 60, 4, 12, 12, 42, 30, 240, 18, 6, 12, 4, 270, 12, 6, 42, 6, 6, 30, 12, 12, 180, 6, 60, 6, 30, 150, 30, 30, 4, 18, 2550, 4, 18, 12, 42, 6, 150, 30, 12, 60, 4, 6, 60, 4, 462, 180, 1230, 18, 30, 108, 60, 180, 12, 6, 30, 6, 570, 420, 462, 180, 6, 4, 198, 42, 522, 600, 1050, 42, 12, 12, 4, 60, 432, 18, 12, 60, 30, 60, 6, 12, 150, 60, 30, 6

Offset: 1

Zhi-Wei Sun, Jun 11 2015

nonn

Conjecture: a(n) exists for any n > 0. Moreover, any positive rational number r can be written as q'/q, where q and q' are terms of A258838 (i.e., q is practical with q-1 and q+1 twin prime, and q' is practical with q'-1 and q'+1 twin prime).
This implies that there are infinitely many "sandwiches of the second kind" (i.e., triples {q-1,q,q+1} with q practical and q-1 and q+1 twin prime).
I have verified the conjecture for all those rational numbers r = n/m with m,n = 1,...,1000. -Zhi-Wei Sun, Jun 15 2015

			a(1) = 4 since 1 = 4/4 with 4 practical and 4-1 and 4+1 twin prime.
a(2) = 6 since 2 = 12/6, 6 is practical with 6-1 and 6+1 twin prime, and 12 is practical with 12-1 and 12+1 twin prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Checking the conjecture for r = n/m with 1 <= n <= m <= 1000
Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A000040, A001359, A006512, A005153, A210479, A258803, A258811, A258838.

f[n_]:=FactorInteger[n]
Pow[n_,i_]:=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
Con[n_]:=Sum[If[Part[Part[f[n],s+1],1]<=DivisorSigma[1,Product[Pow[n,i],{i,1,s}]]+1,0,1],{s,1,Length[f[n]]-1}]
pr[n_]:=n>0&&(n<3||Mod[n,2]+Con[n]==0)
SW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&pr[n]
Do[k=0;Label[bb];k=k+1;If[PrimeQ[Prime[k]+2]&&pr[Prime[k]+1]&&SW[n*(Prime[k]+1)],Goto[aa],Goto[bb]];
Label[aa];Print[n," ",Prime[k]+1];Continue,{n,1,100}]

A259492 Least positive integer k such that prime(k)-k, prime(k)+k, prime(kn)-kn, prime(kn)+kn, prime(k)+kn and prime(kn)+k are all prime.

Original entry on oeis.org

4, 48852, 6, 27330, 89814, 13080, 9570, 44592, 6762, 28560, 1560, 8580, 2958, 672, 9816, 6300, 40050, 53580, 3354, 858, 4530, 100650, 182520, 49740, 48660, 25296, 66990, 87120, 43680, 6840, 52122, 2970, 22770, 15888, 34704, 406350, 67890, 99630, 92490, 83064, 28614, 8580, 32070, 42, 50442, 38676, 818202, 30450, 47880, 4620

Offset: 1

Zhi-Wei Sun, Jun 28 2015

nonn

Conjecture: Any positive rational number r can be written as m/n with prime(m)-m, prime(m)+m, prime(n)-n, prime(n)+n, prime(m)+n and m+prime(n) all prime.

			a(3) = 6 since prime(6)-6 = 7, prime(6)+6 = 19, prime(6*3)-6*3 = 43, prime(6*3)+6*3 = 79, prime(6)+6*3 = 31 and prime(6*3)+6 = 67 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..500
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..150
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A258803, A258836, A259487, A259540.

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

A259531 Least positive integer k such that p(k)^2 + p(k*n)^2 is prime, where p(.) is the partition function given by A000041, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 14, 11, 6, 31, 2, 34, 2, 76, 1, 100, 71, 38, 1, 7, 62, 1128, 1, 180, 123, 15, 174, 128, 4, 111, 110, 2, 4, 2, 2241, 21, 144, 416, 397, 31, 11, 8, 15, 5, 91, 56, 53, 23, 89, 18, 25, 341, 12, 1, 66, 454, 159, 36, 573, 26, 2, 488, 72, 416, 802, 440, 28, 30, 595, 17, 236, 947, 1289, 1287, 1000, 367, 80, 407, 1, 77, 938, 150, 36, 1

Offset: 1

Zhi-Wei Sun, Jul 02 2015

nonn

Conjecture: Any positive rational number r can be written as m/n, where m and n are positive integers with p(m)^2 + p(n)^2 prime.
For example, 4/5 = 124/155, and the number p(124)^2 + p(155)^2 = 2841940500^2 + 66493182097^2 = 4429419891190341567409 is prime.
We also guess that any positive rational number can be written as m/n, where m and n are positive integers with p(m)+p(n) (or p(m)*p(n)-1, or p(m)*p(n)+1) prime.

			a(5) = 6 since p(6)^2 + p(6*5)^2 = 11^2 + 5604^2 = 31404937 is 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..200
Zhi-Wei Sun, Checking the conjecture for 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, A000041, A258803, A258836, A259487, A259492, A259540.

Do[k=0;Label[bb];k=k+1;If[PrimeQ[PartitionsP[k]^2+PartitionsP[k*n]^2],Goto[aa],Goto[bb]];Label[aa];Print[n," ",k]; Continue,{n,1,80}]

A260252 Least prime p such that n = (prime(6q)-1)/(prime(6p)-1) for some prime q.

Original entry on oeis.org

2, 18253, 3, 19, 2, 41, 43, 1087, 263, 29, 2, 281, 83, 8941, 613, 827, 7, 1867, 811, 139, 919, 13, 59, 11551, 10303, 10903, 2707, 3019, 1297, 5, 7333, 1609, 541, 701, 2281, 499, 2713, 6691, 41, 79, 1447, 1409, 263, 2129, 641, 2467, 7741, 1229, 523, 6781

Offset: 1

Zhi-Wei Sun, Jul 20 2015

nonn

Conjecture: Let n be any positive integer, and let s and t belong to the set {1,-1}. Then each positive rational number r can be written as (prime(p*n)+s)/(prime(q*n)+t) with p and q both prime, unless n > r = 1 and {s,t} = {1,-1}.
This extends the conjecture in A258803.
For example, for n = 8, s = t = -1 and r = 16/11, we have (prime(407249*8)-1) /(prime(286411*8)-1) = 54568320/37515720 = r with 407249 and 286411 both prime. Also, for n = 10, s = -1, t = 1, and r = 23/17, we have (prime(1923029*10)-1)/(prime(1444903*10)+1) = 358404768/264907872 = r with 1923029 and 1444903 both prime.

			a(1) = 2 since 1 = (prime(6*2)-1)/(prime(6*2)-1) with 2 prime.
a(2) = 18253 since 2 = 2868672/1434336 = (prime(6*34673)-1)/(prime(6*18253)-1) with 18253 and 34673 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..2000
Zhi-Wei Sun, Checking the conjecture for n = 1..10 and r = a/b with a,b = 1..30
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641.

Cf. A000040, A258803, A260232.

PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/6]
Do[k=0;Label[aa];k=k+1;If[PQ[(Prime[6*Prime[k]]-1)*n+1],Goto[bb],Goto[aa]];Label[bb];Print[n, " ", Prime[k]];Continue,{n,1,50}]

A258811 Least prime p with p+2 prime such that n = (q+1)/(p+1) for some prime q with q+2 also prime.

Original entry on oeis.org

3, 5, 3, 17, 5, 11, 5, 29, 11, 5, 17, 5, 149, 29, 3, 11, 5, 3, 11, 11, 41, 29, 5, 17, 5, 11, 3, 269, 11, 5, 41, 5, 5, 29, 11, 11, 179, 5, 59, 5, 29, 149, 29, 29, 3, 17, 5, 3, 17, 11, 41, 5, 149, 29, 11, 59, 3, 5, 17, 3, 461, 179, 1229, 17, 29, 107, 59, 179, 11, 5

Offset: 1

Zhi-Wei Sun, Jun 11 2015

nonn

Conjecture: a(n) does not exceed n^2-n+5. Also, the set {(p+1)/(q+1): p, q, p+2 and q+2 are all prime} contains all positive rational numbers.

Clearly, this conjecture implies the Twin Prime Conjecture.

			a(1) = 3 since 1 = (3+1)/(3+1) with 3 and 5 twin prime.
a(4) = 17 since 4 = (71+1)/(17+1) with {17,19} and {71,73} twin prime pairs.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A001359, A006512, A258803.

TW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
Do[k=0;Label[bb];k=k+1;If[PrimeQ[Prime[k]+2]&&TW[n*(Prime[k]+1)],Goto[aa],Goto[bb]];
Label[aa];Print[n," ",Prime[k]];Continue,{n,1,70}]

A261295 Least positive integer k such that both k and k*n belong to the set {m>0: prime(m) = prime(p)+2 for some prime p}.

Original entry on oeis.org

3, 3, 6, 578, 18, 3, 6, 90, 1868, 374, 4, 674, 278, 3, 6, 114, 3534, 110, 6, 354, 4, 14, 28464, 2790, 84, 4452, 2802, 3, 6, 3, 90, 2820, 354, 110, 4080, 278, 44, 3, 2712, 18, 3012, 90, 14, 12672, 44, 14, 1572, 1124, 720, 42, 114, 44, 84, 2790, 42, 90, 42, 3, 6, 84, 44, 1572, 3068, 1742, 2394, 174, 110, 744, 3020, 578

Offset: 1

Zhi-Wei Sun, Aug 14 2015

nonn

Conjecture: (i) Any positive rational number r can be written as m/n with m and n in the set S = {k>0: prime(k) = prime(p)+2 for some prime p} = {p+1: p and prime(p)+2 are both prime}.
(ii) Any positive rational number r can be written as m/n with m and n in the set T = {k>0: prime(k) = prime(p)-2 for some prime p} = {p-1: p and prime(p)-2 are both prime}.
(iii) Any positive rational number r not equal to 1 can be written as m/n with m in S and n in T, where the sets S and T are given in parts (i) and (ii).
For example, 4/5 = 15648/19560 with 15647, prime(15647)+2 = 171763, 19559 and prime(19559)+2 = 219409 all prime; and 4/5 = 67536/84420 with 67537, prime(67537)-2 = 848849, 84421 and prime(84421)-2 = 1081937 all prime. Also, 4/5 = 8/10 with 7, prime(7)+2 = 19, 11 and prime(11)-2 = 29 all prime; and 5/4 = 8220/6576 with 8221, prime(8221)+2 = 84349, 6577 and prime(6577)-2 = 65837 all prime.

			a(3) = 6 since prime(6) = 13 = prime(5)+2 with 5 prime, and prime(6*3) = 61 = prime(17)+2 with 17 prime.
a(4) = 578 since prime(578) = 4219 = prime(577)+2 with 577 prime, and prime(578*4) = 20479 = prime(2311)+2 with 2311 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..2000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..200
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A001359, A006512, A218829, A258803, A259487, A259540, A261281, A261282.

f[n_]:=Prime[n]
PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]-2]&&PQ[f[k*n]-2],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]

A261625 Number of primes p <= n such that (p-1)*n+1 is prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 4, 1, 4, 3, 1, 5, 2, 2, 5, 4, 3, 3, 4, 3, 5, 6, 3, 5, 3, 2, 6, 5, 5, 5, 3, 2, 5, 6, 3, 4, 6, 2, 7, 9, 2, 5, 5, 3, 9, 7, 1, 5, 7, 5, 5, 8, 2, 8, 7, 3, 8, 7, 5, 7, 6, 3, 6, 9, 5, 9, 7, 4, 6, 8, 3, 8, 9, 3

Offset: 1

Zhi-Wei Sun, Aug 27 2015

nonn

Conjecture: a(n) > 0 for all n > 1.

			a(53) = 1 since 3 and (3-1)*53+1 = 107 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2015.

Cf. A000040, A034693, A258803, A258811, A261437.

Do[r=0;Do[If[PrimeQ[(Prime[k]-1)n+1],r=r+1],{k,1,PrimePi[n]}];Print[n," ",r];Continue,{n,1,80}]

a(n) = my(nb=0); forprime(p=2, n, if (isprime((p-1)*n+1), nb++)); nb; \\ Michel Marcus, Aug 27 2015

cp's OEIS Frontend

A259487 Least positive integer m with prime(m)+2 and prime(prime(m))+2 both prime such that prime(m*n)+2 and prime(prime(m*n))+2 are both prime.

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A259540 Least positive integer k such that k and k*n are terms of A259539.

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A258836 Least practical number q with q-1 and q+1 twin prime such that n = q'/q for some practical number q' with q'-1 and q'+1 twin prime.

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A259492 Least positive integer k such that prime(k)-k, prime(k)+k, prime(k*n)-k*n, prime(k*n)+k*n, prime(k)+k*n and prime(k*n)+k are all prime.

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A259531 Least positive integer k such that p(k)^2 + p(k*n)^2 is prime, where p(.) is the partition function given by A000041, or 0 if no such k exists.

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A260252 Least prime p such that n = (prime(6*q)-1)/(prime(6*p)-1) for some prime q.

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A258811 Least prime p with p+2 prime such that n = (q+1)/(p+1) for some prime q with q+2 also prime.

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A261295 Least positive integer k such that both k and k*n belong to the set {m>0: prime(m) = prime(p)+2 for some prime p}.

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A259487 Least positive integer m with prime(m)+2 and prime(prime(m))+2 both prime such that prime(mn)+2 and prime(prime(mn))+2 are both prime.

A259492 Least positive integer k such that prime(k)-k, prime(k)+k, prime(kn)-kn, prime(kn)+kn, prime(k)+kn and prime(kn)+k are all prime.

A260252 Least prime p such that n = (prime(6q)-1)/(prime(6p)-1) for some prime q.