A258836 -id:A258836 - 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}]

A258838 Practical numbers q with q-1 and q+1 twin primes: "Sandwiches of the second kind".

Original entry on oeis.org

4, 6, 12, 18, 30, 42, 60, 72, 108, 150, 180, 192, 198, 228, 240, 270, 312, 348, 420, 432, 462, 522, 570, 600, 660, 810, 828, 858, 882, 1020, 1032, 1050, 1092, 1152, 1230, 1290, 1302, 1320, 1428, 1452, 1482, 1488, 1620, 1722, 1872, 1932, 1950, 1998, 2028, 2088, 2112, 2130, 2142, 2268, 2310, 2340, 2550, 2592, 2688, 2730

Offset: 1

Zhi-Wei Sun, Jun 12 2015

nonn

The author introduced two kinds of "sandwiches" in 2013. The conjecture in A258836 essentially says that {a(m)/a(n): m,n = 1,2,3,...} coincides with the set of all positive rational numbers. This implies that the sequence contains infinitely many terms.

			a(1) = 4 since 4 is practical with 4-1 and 4+1 twin prime.
a(2) = 6 since 6 is practical with 6-1 and 6+1 twin prime.
a(3) = 12 since 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, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A001359, A005153, A006512, A014574, A209236, A210452, A210479, A258836.

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]
n=0;Do[If[SW[m],n=n+1;Print[n," ",m]],{m,1,2730}]

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}]

A257922 Practical numbers m with m-1 and m+1 both prime, and prime(m)-1 and prime(m)+1 both practical.

Original entry on oeis.org

4, 522, 1932, 5100, 6132, 6552, 8220, 18312, 18540, 22110, 29568, 45342, 70488, 70950, 92220, 105360, 109662, 114600, 116532, 117192, 123552, 128982, 131838, 132762, 136710, 148302, 149160, 166848, 177012, 183438, 197340, 206280, 233550, 235008, 257868, 272808, 273900, 276780, 279708, 286590

Offset: 1

Zhi-Wei Sun, Jul 12 2015

nonn

Conjecture: The sequence contains infinitely many terms. In other words, there are infinitely many positive integers n such that {prime(n)-1, prime(n), prime(n)+1} is a "sandwich of the first kind" (A210479) and {n-1, n, n+1} is a "sandwich of the second kind" (A258838).

This implies that there are infinitely many sandwiches of the first kind and also there are infinitely many sandwiches of the second kind.

			a(1) = 4 since 4 is paractical with 4-1 and 4+1 twin prime, and prime(4)-1 = 6 and prime(4)+1 = 8 are both practical.
a(2) = 522 since 522 is paractical with 522-1 and 522+1 twin prime, and prime(522)-1 = 3738 and prime(522)+1 = 3740 are both practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
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, A005153, A014574, A209236, A210479, A257924, A258836, A258838, A259539.

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)
n=0;Do[If[PrimeQ[Prime[k]+2]&&pr[Prime[k]+1]&&pr[Prime[Prime[k]+1]-1]&&pr[Prime[Prime[k]+1]+1],n=n+1;Print[n," ",Prime[k]+1]],{k,1,24962}]

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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A258838 Practical numbers q with q-1 and q+1 twin primes: "Sandwiches of the second kind".

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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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A257922 Practical numbers m with m-1 and m+1 both prime, and prime(m)-1 and prime(m)+1 both practical.

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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.