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 1-6 of 6 results.

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

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 *)

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 *)

A153413 Smaller of twin prime pairs such that p1*p2+average_of_twin_prime_pair=prime.

Original entry on oeis.org

3, 5, 29, 59, 137, 179, 239, 419, 617, 1049, 1607, 1697, 1787, 2267, 2309, 2729, 3257, 3389, 3527, 3767, 4157, 4217, 4337, 4799, 5639, 5867, 6659, 6689, 6869, 6959, 7487, 7547, 7589, 8537, 8627, 8969, 9629, 9857, 9929, 10457, 11117, 11969, 12539, 13337
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Comments

3*5+4=19 prime, 5*7+6=41 prime, 29*31+30=929 prime, ...

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];If[p2-p1==2,a=p1*p2+(p1+1);If[PrimeQ[a],AppendTo[lst,p1]]],{n,7!}];lst

A153414 Larger of twin prime pairs such that p1*p2+average_of_twin_prime_pair=prime.

Original entry on oeis.org

5, 7, 31, 61, 139, 181, 241, 421, 619, 1051, 1609, 1699, 1789, 2269, 2311, 2731, 3259, 3391, 3529, 3769, 4159, 4219, 4339, 4801, 5641, 5869, 6661, 6691, 6871, 6961, 7489, 7549, 7591, 8539, 8629, 8971, 9631, 9859, 9931, 10459, 11119, 11971, 12541, 13339
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Keywords

Comments

3*5+4=19 prime, 5*7+6=41 prime, 29*31+30=929 prime, ...

Links

Crossrefs

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153406, A153407, A153408, A153409, A153410, A153413

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];If[p2-p1==2,a=p1*p2+(p1+1);If[PrimeQ[a],AppendTo[lst,p2]]],{n,7!}];lst
Transpose[Select[Select[Partition[Prime[Range[1600]],2,1],Last[#]- First[#] == 2&], PrimeQ[Times@@#+Mean[#]]&]][[2]] (* Harvey P. Dale, Jan 23 2012 *)
Showing 1-6 of 6 results.

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