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 12 results. Next

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

A153404 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

5, 571, 1753, 5113, 6949, 9283, 11047, 14401, 24859, 25171, 26203, 31159, 34471, 41719, 42397, 45289, 61099, 62533, 80611, 82141, 90001, 91969, 92347, 93811, 98377, 98887, 103591, 105907, 111373, 117133, 120997, 122827, 128413, 135607
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Examples

			3*5*7 + 2 + 2 - 1 = 108 and 108 +- 1 are primes.

Links

Crossrefs

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

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

A153409 Smallest of 3 consecutive prime numbers such that p1*p2*p3*d1*d2=average of twin prime pairs; p1,p2,p3 consecutive prime numbers; d1(delta)=p2-p1, d2(delta)=p3-p2.

Original entry on oeis.org

2, 3, 19, 61, 229, 499, 677, 1009, 1753, 2089, 2791, 3167, 10657, 12379, 12893, 13477, 15139, 18553, 20551, 21871, 25367, 26227, 26669, 33601, 36781, 36931, 41399, 41413, 43543, 61543, 63331, 63839, 68903, 71993, 75709, 76343, 76471, 86629
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Comments

2*3*5*1*2=60+-1=primes, 3*5*7*2*2=420+-1=primes, 19*23*29*4*6=304152+-1=primes,...

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153406, A153407, A153408

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;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p1]],{n,8!}];lst
cpnQ[{a_,b_,c_}]:=Module[{pr=a*b*c*(b-a)*(c-b)},AllTrue[pr+{1,-1}, PrimeQ]]; Transpose[Select[Partition[Prime[Range[10000]],3,1], cpnQ]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 24 2015 *)

A153405 Larger 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

7, 577, 1759, 5119, 6959, 9293, 11057, 14407, 24877, 25183, 26209, 31177, 34483, 41729, 42403, 45293, 61121, 62539, 80621, 82153, 90007, 91997, 92353, 93827, 98387, 98893, 103613, 105913, 111409, 117163, 121001, 122833, 128431, 135613
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Examples

			7 is a term since (3, 5, 7) are consecutive primes, 3*5*7 + 2 + 2 - 1 = 108, and 108 +-1 = are twin primes.

Links

Crossrefs

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

Programs

  • Magma
    [p3:k in [1..14000]| IsPrime(p1*p2*p3+p3-p1-2) and IsPrime(p1*p2*p3+p3-p1) where p1 is NthPrime(k) where p2 is NthPrime(k+1) where p3 is NthPrime(k+2) ]; // Marius A. Burtea, Dec 31 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 (* Vladimir Joseph Stephan Orlovsky *)
okQ[{a_, b_, c_}] := Module[{x = a b c + (b - a) + (c - b) - 1}, PrimeQ[x - 1] && PrimeQ[x + 1]]
Transpose[Select[Partition[Prime[Range[15000]], 3, 1], okQ]][[3]] (* Harvey P. Dale, Jan 18 2011 *)

A153410 Middle of 3 consecutive prime numbers, p1, p2, p3, such that p1*p2*p3*d1*d2 = average of twin prime pairs; d1 (delta) = p2 - p1, d2 (delta) = p3 - p2.

Original entry on oeis.org

3, 5, 23, 67, 233, 503, 683, 1013, 1759, 2099, 2797, 3169, 10663, 12391, 12899, 13487, 15149, 18583, 20563, 21881, 25373, 26237, 26681, 33613, 36787, 36943, 41411, 41443, 43573, 61547, 63337, 63841, 68909, 71999, 75721, 76367, 76481, 86677
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Examples

			2*3*5*1*2 = 60 and 60 +- 1 are primes.
3*5*7*2*2 = 420 and 420 +- 1 are primes.
19*23*29*4*6 = 304152 and 304152 +- 1 are primes.

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153406, A153407, A153408, A153409.

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;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p2]],{n,8!}];lst
cpnQ[{a_,b_,c_}]:=Module[{x=Times@@Join[{a,b,c},Differences[ {a,b,c}]]}, AllTrue[ x+{1,-1},PrimeQ]]; Select[Partition[ Prime[Range[ 10000]],3,1], cpnQ][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 01 2020 *)

A153411 Larger of 3 consecutive prime numbers such that p1*p2*p3*d1*d2=average of twin prime pairs; p1,p2,p3 consecutive prime numbers; d1(delta)=p2-p1, d2(delta)=p3-p2.

Original entry on oeis.org

5, 7, 29, 71, 239, 509, 691, 1019, 1777, 2111, 2801, 3181, 10667, 12401, 12907, 13499, 15161, 18587, 20593, 21893, 25391, 26249, 26683, 33617, 36791, 36947, 41413, 41453, 43577, 61553, 63347, 63853, 68917, 72019, 75731, 76369, 76487, 86689
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Comments

2*3*5*1*2=60+-1=primes, 3*5*7*2*2=420+-1=primes, 19*23*29*4*6=304152+-1=primes,...

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153406, A153407, A153408, A153409, A153410

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;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p3]],{n,8!}];lst
tppQ[n_]:=Module[{c=Times@@Join[n,Differences[n]]},AllTrue[c+{1,-1}, PrimeQ]]; Transpose[Select[Partition[Prime[Range[10^4]],3,1], tppQ]] [[3]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 17 2016 *)
Showing 1-10 of 12 results. Next

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