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.

Previous Showing 11-20 of 20 results.

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 *)

A178421 Lower primes p1 in a twin pair such that sum of p1 and p2 yields average a1 of twin prime pairs and product of 2*a1 is another average of twin prime pairs.

Original entry on oeis.org

211049, 248639, 253679, 410339, 507359, 605639, 1121189, 1138829, 1262099, 2162579, 2172869, 2277659, 4070219, 6305459, 7671509, 11659409, 12577109, 14203769, 14862119, 17472839, 18728639, 18798359, 20520569, 21140699
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 27 2010

Keywords

Comments

The definition means that a1/2, a1 and 2*a1 are all in A014574 (twin prime averages). - R. J. Mathar, Nov 02 2023

Examples

			211049 is a term since 211049 and 211051 are twin primes; 211049 + 211051 = 422100 is an average of twin primes, and 2*422100 = 844200 is another average of twin primes.

Links

Crossrefs

Cf. A069175, A014574, A069142, A099349.

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];a1=p1+p2;a2=2*a1;If[p2-p1==2&&PrimeQ[a1-1]&&PrimeQ[a1+1]&&PrimeQ[a2-1]&&PrimeQ[a2+1],AppendTo[lst,p1]],{n,10!}];lst
atpQ[{a_,b_}]:=Module[{m=a+b},b-a==2&&AllTrue[m+{1,-1},PrimeQ] && AllTrue[ 2m+{1,-1},PrimeQ]]; Select[Partition[Prime[Range[134*10^4]],2,1],atpQ][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 28 2019 *)

Formula

a(n) = A069175(n)-1. - R. J. Mathar, Nov 02 2023

A158350 Primes p such that previousPrime(p) + p -+ 1 are twin primes.

Original entry on oeis.org

7, 11, 17, 23, 31, 71, 101, 127, 233, 307, 311, 409, 419, 443, 617, 647, 661, 719, 743, 811, 839, 863, 941, 1049, 1061, 1361, 1487, 1667, 1697, 1889, 2003, 2053, 2129, 2131, 2243, 2267, 2551, 2647, 2711, 2753, 2767, 2833, 3049, 3109, 3163, 3229, 3299, 3331
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 16 2009

Keywords

Comments

Sum of two consecutive primes = arithmetic mean of twin primes.

Examples

			5+7=12-+1 primes, 7+11=18-+1 primes, 13+17-+1 primes, ...

Links

Crossrefs

Cf. A099349.

Programs

  • Maple
    t1:=[]; for n from 2 to 1000 do p:=ithprime(n); q:=prevprime(p);
if isprime(p+q-1) and isprime(p+q+1) then t1:=[op(t1),p]; fi; od: t1; # N. J. A. Sloane, Dec 24 2012
  • Mathematica
    lst={}; Do[p0=Prime[n]; p1=Prime[n+1]; a=p0+p1; If[PrimeQ[a-1] && PrimeQ[a+1], AppendTo[lst, p1]], {n, 1000}]; lst

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 *)

A158351 Primes p0 such that p0+p1+p2-+2 are primes; p0,p1,p2 are three consecutive primes.

Original entry on oeis.org

3, 251, 523, 1063, 4007, 4373, 4423, 7517, 11801, 11833, 11927, 12491, 12757, 12967, 15817, 15907, 16381, 16481, 16763, 16987, 17851, 21341, 21937, 22343, 22441, 22877, 23327, 25849, 26591, 26993, 27061, 31153, 31321, 31583, 33773, 35159
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 16 2009

Keywords

Comments

sum of three consecutive primes = arithmetical mean of two primes. 3+5+7=15-+2 (13,17)primes, 251+257+263-+2 (769,773)primes, ...

Links

Crossrefs

Cf. A099349, A158350.

Programs

  • Mathematica
    lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];a=p0+p1+p2;If[PrimeQ[a-2]&&PrimeQ[a+2],AppendTo[lst,p0]],{n,2*7!}];lst
Select[Partition[Prime[Range[4000]],3,1],AllTrue[Total[#]+{2,-2},PrimeQ]&][[;;,1]] (* Harvey P. Dale, Apr 23 2024 *)
  • PARI
    is(n)=my(p=nextprime(n+1),q=nextprime(p+1)); isprime(n) && isprime(n+p+q-2) && isprime(n+p+q+2) \\ Charles R Greathouse IV, Jan 29 2016
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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