A259539 -id:A259539 - cp's OEIS frontend

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

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}]

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}]

A257924 Primes p with p-1, p+1, prime(p)-1 and prime(p)+1 all practical.

Original entry on oeis.org

3, 7, 31, 89, 199, 8009, 11551, 20129, 23549, 38609, 47501, 67231, 96221, 97001, 103409, 111871, 120473, 131071, 143261, 146681, 168869, 174761, 183091, 193951, 196181, 208279, 208961, 219727, 229769, 237691, 238519, 240641, 247759, 270271, 290249, 291101, 293201, 337039, 340577, 352831

Offset: 1

Zhi-Wei Sun, Jul 13 2015

nonn

Conjecture: The sequence contains infinitely many terms. In other words, there are infinitely many prime numbers p such that {p-1, p, p+1} and {prime(p)-1, prime(p), prime(p)+1} are both "sandwiches of the first kind" (A210479).

			a(1) = 3 since 3 is prime with 3-1, 3+1, prime(3)-1 = 4 and prime(3)+1 = 6 all practical.
a(3) = 31 since 31 is prime with 31-1, 31+1, prime(31)-1 = 126 and prime(31)+1 = 128 all practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
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, A209236, A210479, A257922, 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[pr[Prime[k]-1]&&pr[Prime[k]+1]&&pr[Prime[Prime[k]]-1]&&pr[Prime[Prime[k]]+1],n=n+1;Print[n," ",Prime[k]]],{k,1,30201}]

A261533 Primes p such that p+2 is prime with prime(p+2)-prime(p)=6.

Original entry on oeis.org

3, 5, 59, 2789, 5231, 6947, 8087, 11717, 15269, 16229, 17207, 17909, 18059, 18131, 24917, 28751, 35279, 37307, 39227, 39239, 41201, 43787, 45821, 47741, 51869, 53087, 53609, 58439, 64577, 69857, 70919, 75707, 79631, 84869, 92381, 93479, 96179, 102197, 102929, 106187

Offset: 1

Zhi-Wei Sun, Aug 23 2015

nonn

The conjecture in A261528 implies that the current sequence has infinitely many terms.

Note that for each n > 2 the difference prime(n+2)-prime(n) is at least 6.

			a(1) = 3 since 3 and 3+2 = 5 are twin prime, and prime(5)-prime(3) = 11-5 = 6.
a(2) = 5 since 5 and 5+2 = 7 are twin prime, and prime(7)-prime(5) = 17-11 = 6.

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, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A001359, A006512, A023201, A046117, A236458, A259488, A259539, A261528.

f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6
n=0;Do[If[PQ[k],n=n+1;Print[n," ",f[k]]],{k,1,10119}]
Select[Partition[Prime[Range[11000]],2,1],#[[2]]-#[[1]]==2&&Prime[#[[1]]+ 2]- Prime[#[[1]]]==6&][[All,1]] (* Harvey P. Dale, Apr 26 2020 *)

isok(i)=p=prime(i);isprime(p+2)&&prime(p+2)-prime(p)==6;
first(m)=my(v=vector(m));i=1;for(j=1,m,while(!isok(i),i++);v[j]=prime(i);i++);v; \\ Anders Hellström, Aug 23 2015

A259628 Numbers m with m-1, m+1, prime(m)+2, prime(m)-m, prime(m)+m, mprime(m)-1 and mprime(m)+1 all prime.

Original entry on oeis.org

2523708, 6740478, 6759030, 14655522, 22885698, 28384200, 44630148, 71742300, 87002328, 87466500, 89842200, 153110622, 153647490, 184373490, 283232040, 312124920, 366318960, 408770670, 412216920, 439429128, 456486030, 486730398, 517602600, 606159978, 607942848, 675661080, 855983352, 869593998, 923864562, 971400672

Offset: 1

Zhi-Wei Sun, Jul 01 2015

nonn

Conjecture: The sequence contains infinitely many terms.

This is stronger than the conjectures in A232861 and A259539.

			a(1) = 2523708 since the seven numbers 2523707, 2523709, prime(2523708)+2 = 41578739+2 = 41578741, prime(2523708)-2523708 = 41578739-2523708 = 39055031, prime(2523708)+2523708 = 41578739+2523708 = 44102447, 2523708*prime(2523708)-1 = 2523708*41578739-1 = 104932596244211 and 2523708*prime(2523708)+1 = 2523708*41578739+1 = 104932596244213 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..160
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A014574, A232861, A259539.

TW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
n=0;Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]+1]+2]&&PrimeQ[Prime[Prime[k]+1]-Prime[k]-1]&&PrimeQ[Prime[Prime[k]+1]+Prime[k]+1]&&TW[(Prime[k]+1)Prime[Prime[k]+1]],n=n+1;Print[n," ",Prime[k]+1]],{k,1,5*10^7}]
allprQ[n_]:=Module[{p=Prime[n]},AllTrue[{n-1,n+1,p+2,p-n,p+n,n*p-1, n*p+1}, PrimeQ]]; Select[Range[98*10^7],allprQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 26 2016 *)

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

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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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A257924 Primes p with p-1, p+1, prime(p)-1 and prime(p)+1 all practical.

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A261533 Primes p such that p+2 is prime with prime(p+2)-prime(p)=6.

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A259628 Numbers m with m-1, m+1, prime(m)+2, prime(m)-m, prime(m)+m, m*prime(m)-1 and m*prime(m)+1 all prime.

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A259628 Numbers m with m-1, m+1, prime(m)+2, prime(m)-m, prime(m)+m, mprime(m)-1 and mprime(m)+1 all prime.