cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 20 results. Next

A153374 Smaller of two consecutive prime numbers such that p0+p1=average of twin prime pairs and p0*p1+7=average of twin prime pairs.

Original entry on oeis.org

5, 13, 1039, 2753, 3343, 22381, 45979, 88223, 92317, 135221, 154153, 233323, 287149, 344221, 365293, 392723, 479629, 549739, 574363, 650581, 659423, 666079, 749803, 786949, 869059, 959723, 1023521, 1045027, 1161841, 1180423, 1193021
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 24 2008

Keywords

Comments

5+7=12+-1=primes, 5*7+7=42+-1=primes; 13+17=30+-1=primes, 13*17+7=228+-1=primes;...

Crossrefs

Cf. A099349

Programs

  • Mathematica
    lst={};Do[p0=Prime[n];p1=Prime[n+1];a=p0+p1;b=p0*p1+7;If[PrimeQ[a-1]&&PrimeQ[a+1]&&PrimeQ[b-1]&&PrimeQ[b+1],AppendTo[lst,p0]],{n,9!}];lst

A153375 Larger of two consecutive prime numbers such that p0+p1=average of twin prime pairs and p0*p1+7=average of twin prime pairs.

Original entry on oeis.org

7, 17, 1049, 2767, 3347, 22391, 45989, 88237, 92333, 135241, 154157, 233327, 287159, 344231, 365297, 392737, 479639, 549749, 574367, 650591, 659437, 666089, 749807, 786959, 869069, 959737, 1023541, 1045043, 1161851, 1180427, 1193041
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 24 2008

Keywords

Comments

5+7=12+-1=primes, 5*7+7=42+-1=primes; 13+17=30+-1=primes, 13*17+7=228+-1=primes;...

Links

Crossrefs

Cf. A099349, A153374

Programs

  • Mathematica
    lst={};Do[p0=Prime[n];p1=Prime[n+1];a=p0+p1;b=p0*p1+7;If[PrimeQ[a-1]&&PrimeQ[a+1]&&PrimeQ[b-1]&&PrimeQ[b+1],AppendTo[lst,p1]],{n,9!}];lst
atpQ[{a_,b_}]:=Module[{t=a+b,p=a*b},AllTrue[{t-1,t+1,p+6,p+8},PrimeQ]]; Transpose[ Select[Partition[Prime[Range[100000]],2,1],atpQ]][[2]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 09 2014 *)

A153376 Smaller of two consecutive prime numbers such that p1*p2*d + d = average of twin prime pairs, d (delta) = p2 - p1.

Original entry on oeis.org

5, 7, 41, 43, 101, 103, 113, 227, 331, 379, 569, 647, 733, 751, 863, 971, 1093, 1123, 1163, 1217, 1381, 1499, 2063, 2131, 2179, 2311, 2357, 2399, 2707, 2711, 3709, 4789, 4817, 5021, 5051, 5171, 5479, 5501, 5987, 6011, 6827, 6949, 6967, 7103, 7213, 7477
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 24 2008

Keywords

Comments

See A153377 for the corresponding larger prime.

Examples

			5*7*2 + 2 = 72 and 72 +- 1 are primes, so 5 is a term.
7*11*4 + 4 = 312 and 312 +- 1 are primes, so 7 is a term.

Links

Crossrefs

Cf. A099349, A153374, A153375, A153377.

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];d=p2-p1;a=p1*p2*d+d;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p1]],{n,7!}];lst

A153377 Larger of two consecutive prime numbers such that p1*p2*d + d = average of twin prime pairs, d (delta) = p2 - p1.

Original entry on oeis.org

7, 11, 43, 47, 103, 107, 127, 229, 337, 383, 571, 653, 739, 757, 877, 977, 1097, 1129, 1171, 1223, 1399, 1511, 2069, 2137, 2203, 2333, 2371, 2411, 2711, 2713, 3719, 4793, 4831, 5023, 5059, 5179, 5483, 5503, 6007, 6029, 6829, 6959, 6971, 7109, 7219, 7481
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 24 2008

Keywords

Comments

See A153376 for the corresponding lesser prime.

Examples

			5*7*2 + 2 = 72 and 72 +- 1 are primes, so 7 is a term.
7*11*4 + 4 = 312 and 312 +- 1 are primes, so 11 is a term.

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376.

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];d=p2-p1;a=p1*p2*d+d;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p2]],{n,7!}];lst

A153378 Smaller of two consecutive prime numbers such that p1*p2*d - d = average of twin prime pairs, d (delta) = p2 - p1.

Original entry on oeis.org

1187, 8893, 13967, 31817, 33107, 56009, 57587, 66587, 85837, 87797, 90547, 91199, 93497, 101293, 103177, 111667, 113117, 127447, 141397, 142873, 150343, 150407, 151667, 152617, 156817, 157127, 161977, 176819, 179737, 186007, 205957, 209401
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 24 2008

Keywords

Comments

See A153379 for the corresponding larger prime.

Examples

			1187*1193*6 - 6 = 8496540 and 8496540 +- 1 are primes, so 1187 is a term.

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376, A153377, A153379.

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];d=p2-p1;a=p1*p2*d-d;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p1]],{n,8!}];lst

A153379 Larger of two consecutive prime numbers, p1 and p2 = p1 + d, such that p1*p2*d - d is the average of twin primes.

Original entry on oeis.org

1193, 8923, 13997, 31847, 33113, 56039, 57593, 66593, 85843, 87803, 90583, 91229, 93503, 101323, 103183, 111697, 113123, 127453, 141403, 142897, 150373, 150413, 151673, 152623, 156823, 157133, 161983, 176849, 179743, 186013, 205963, 209431
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 24 2008

Keywords

Examples

			1193 since 1187 and 1193 = 1187 + 6 are consecutive primes, 1187*1193*6 - 6 = 8496540, and (8496539, 8496541) are twin primes.

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376, A153377, A153378.

Programs

  • Magma
    [q:p in PrimesUpTo(210000)| IsPrime(a-1) and IsPrime(a+1) where a is (p*q-1)*(q-p) where q is NextPrime(p)]; // Marius A. Burtea, Jan 03 2020
  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];d=p2-p1;a=p1*p2*d-d;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p2]],{n,8!}];lst
l2cpQ[{a_,b_}]:=Module[{d=b-a},AllTrue[a*b*d-d+{1,-1},PrimeQ]]; Transpose[ Select[ Partition[Prime[Range[20000]],2,1],l2cpQ]][[2]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 18 2015 *)

Extensions

Name edited by Amiram Eldar, Jan 03 2020

A153406 Smallest of 3 consecutive prime numbers such that p1*p2*p3+d1+d2+1=average of twin prime pairs, d1(delta)=p2-p1,d2(delta)=p3-p2.

Original entry on oeis.org

4813, 9007, 13831, 33791, 35023, 48337, 51577, 52153, 61297, 62207, 77743, 95107, 102607, 105137, 105673, 109663, 111767, 114781, 119747, 128221, 135367, 136727, 138679, 149197, 153949, 159787, 163199, 165437, 174829, 188677, 195973, 208009
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Comments

4813*4817*4831+4+14=112002971670+-1=primes,...

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153402, A153404, A153405

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];p3=Prime[n+2];d1=p2-p1;d2=p3-p2;a=p1*p2*p3+d1+d2+1;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p1]],{n,8!}];lst
s3cpnQ[n_]:=Module[{c=Times@@n+Total[Differences[n]]+1},AllTrue[c+{1,-1}, PrimeQ]]; Transpose[Select[Partition[ Prime[Range[ 20000]],3,1], s3cpnQ]] [[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 05 2014 *)

A153407 Middle of 3 consecutive prime numbers such that p1*p2*p3+d1+d2+1=average of twin prime pairs, d1(delta)=p2-p1,d2(delta)=p3-p2.

Original entry on oeis.org

4817, 9011, 13841, 33797, 35027, 48341, 51581, 52163, 61331, 62213, 77747, 95111, 102611, 105143, 105683, 109673, 111773, 114797, 119759, 128237, 135389, 136733, 138683, 149213, 153953, 159791, 163211, 165443, 174851, 188681, 195977, 208037
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Comments

4813*4817*4831+4+14=112002971670+-1=primes,...

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153402, A153404, A153405, A153406

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];p3=Prime[n+2];d1=p2-p1;d2=p3-p2;a=p1*p2*p3+d1+d2+1;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p2]],{n,8!}];lst
cnpQ[{a_,b_,c_}]:=Module[{p=a*b*c+(b-a)+(c-b)+1},And@@PrimeQ[p+{1,-1}]]; Transpose[Select[Partition[Prime[Range[20000]],3,1],cpnQ]][[2]] (* Harvey P. Dale, Jul 30 2013 *)

A153402 Smaller of 3 consecutive prime numbers such that p1*p2*p3+d1+d2-1=average of twin prime pairs, d1(delta)=p2-p1,d2(delta)=p3-p2.

Original entry on oeis.org

3, 569, 1747, 5107, 6947, 9281, 11027, 14389, 24851, 25169, 26189, 31153, 34469, 41687, 42391, 45281, 61091, 62507, 80603, 82139, 89989, 91967, 92333, 93809, 98369, 98873, 103583, 105899, 111347, 117127, 120977, 122819, 128411, 135601
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Comments

3*5*7+2+2-1=108+-1=prime,

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];p3=Prime[n+2];d1=p2-p1;d2=p3-p2;a=p1*p2*p3+d1+d2-1;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p1]],{n,8!}];lst
s3cpQ[{a_,b_,c_}]:=Module[{tp=a*b*c+(c-a)-1},AllTrue[tp+{1,-1},PrimeQ]]; Select[ Partition[Prime[Range[15000]],3,1],s3cpQ][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 01 2018 *)

A153408 Largest of 3 consecutive prime numbers such that p1*p2*p3 + d1 + d2 + 1 = average of twin prime pairs, d1 (delta) = p2 - p1, d2 (delta) = p3 - p2.

Original entry on oeis.org

4831, 9013, 13859, 33809, 35051, 48353, 51593, 52177, 61333, 62219, 77761, 95131, 102643, 105167, 105691, 109717, 111779, 114799, 119771, 128239, 135391, 136739, 138727, 149239, 153991, 159793, 163223, 165449, 174859, 188687, 195991, 208049
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Examples

			4813*4817*4831 + 4 + 14 = 112002971670 and 112002971670 +- 1 are primes.

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153402, A153404, A153405, A153406, A153407.

Programs

  • Magma
    [NthPrime(k+2):k in [1..20000]| IsPrime(q-1) and IsPrime(q+1) where q is NthPrime(k)* NthPrime(k+1)* NthPrime(k+2)+ NthPrime(k+2)- NthPrime(k)+1]; // Marius A. Burtea, Dec 22 2019
  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];p3=Prime[n+2];d1=p2-p1;d2=p3-p2;a=p1*p2*p3+d1+d2+1;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p3]],{n,8!}];lst
Select[Partition[Prime[Range[20000]],3,1],AllTrue[Times@@#+Total[ Differences[ #]]+ {2,0},PrimeQ]&][[All,3]] (* Harvey P. Dale, Apr 22 2022 *)

Extensions

Definition modified by Harvey P. Dale, Apr 22 2022
Showing 1-10 of 20 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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