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 1-9 of 9 results.

A242772 The lesser of twin primes p1 such that 2*p1 + p2 is a prime number (A174913) and also the lesser of other twin primes in A174913.

Original entry on oeis.org

5, 11489, 32969, 33329, 33599, 42839, 58109, 93809, 96329, 114599, 180179, 272999, 309539, 334889, 401309, 540539, 633569, 717089, 784349, 820409, 870239, 879689, 907139, 948089, 989249, 991619, 994559, 1020959, 1028579, 1044749, 1185659, 1189649, 1245449, 1253909
Offset: 1

Views

Author

Ivan N. Ianakiev, May 22 2014

Keywords

Comments

It seems that a(n) == 9 mod 10 for n > 1.
a(n) == 9 (mod 10) for n > 1 since if p1 == 1, 3 or 7 (mod 10) then 2*p1 + p2, p2, or 2*p1 + p2 + 2 is divisible by 5, respectively. - Amiram Eldar, Dec 31 2019

Examples

			a(1) = A174913(2) = 5 and 2*5 + 7 = 17 = A174913(3).

Links

Crossrefs

Cf. A001359, A174913, A174920, A242773.

Programs

  • Mathematica
    Select[Range[10^6], And @@ PrimeQ[{#, # + 2,(p = 3*# + 2), p + 2, 3*p + 2}] &] (* Amiram Eldar, Dec 31 2019 *)
  • PARI
    isok(p) = isprime(p) && isprime(p+2) && isprime(q=3*p+2) && isprime(q+2) && isprime(3*q+2); \\ Michel Marcus, May 23 2014

A242773 The greater of twin primes p2 such that 2*p1 + p2 is a prime number (A174913) and also the lesser of other twin primes in A174913.

Original entry on oeis.org

7, 11491, 32971, 33331, 33601, 42841, 58111, 93811, 96331, 114601, 180181, 273001, 309541, 334891, 401311, 540541, 633571, 717091, 784351, 820411, 870241, 879691, 907141, 948091, 989251, 991621, 994561, 1020961, 1028581, 1044751, 1185661, 1189651, 1245451, 1253911
Offset: 1

Views

Author

Ivan N. Ianakiev, May 22 2014

Keywords

Comments

It seems that a(n) == 1 mod 10 for n > 1.
a(n) == 1 (mod 10) for n > 1 since if p2 == 3, 7 or 9 (mod 10) then 2*p1 + p2, p1, or 2*p1 + p2 + 2 is divisible by 5, respectively. - Amiram Eldar, Dec 31 2019

Examples

			a(1) = 7, 7 - 2 = 5 = A174913(1) and 2*A174913(1) + 7 = A174913(2).

Links

Crossrefs

Cf. A001359, A174913, A242772.

Programs

  • Mathematica
    Select[Range[10^6], And @@ PrimeQ[{#, # + 2, (p = 3*# + 2), p + 2, 3*p + 2}] &] + 2 (* Amiram Eldar, Dec 31 2019 *)

Formula

a(n) = A242772(n) + 2.

A174920 List of primes p1 such that (p1,p2) are twin primes where both 2*p1+p2 and p1+2*p2 are primes.

Original entry on oeis.org

3, 5, 59, 269, 1949, 2999, 6359, 11489, 11549, 14549, 18539, 19889, 21839, 31079, 32909, 32969, 33329, 33599, 42569, 42839, 50459, 53549, 58109, 68879, 70199, 74609, 79229, 80909, 93809, 96329, 98909, 104309, 109139, 114599, 121019, 125789
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 02 2010

Keywords

Comments

Terms >5 are congruent to 29 mod 30. - Zak Seidov, May 10 2012
Also 2*p1+p2 and p1+2*p2 are twin primes. - Zak Seidov, May 10 2012

Examples

			a(1)=3 because 3, 5 are twin primes and 2*3+5=11, 3+2*5=13 are also primes.

Links

Crossrefs

Cf. A001359, A174913, A174915, A174916, A174917, A177335.
Cf. A132929, A177336.

Programs

  • Magma
    [NthPrime(n): n in [1..12000] | forall{p: p in [NthPrime(n)+2,3*NthPrime(n)+2,3*NthPrime(n)+4] | IsPrime(p)}]; // Bruno Berselli, May 10 2012
  • Maple
    select(q -> isprime(q) and isprime(q+2) and isprime(3*q+2) and isprime(3*q+4), [3,5,seq(i,i=29..200000,30)]); # Robert Israel, May 06 2019
  • Mathematica
    lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[2*p1+p2]&&PrimeQ[p1+2*p2],AppendTo[lst,p1]],{n,8!}];lst

Formula

From Wesley Ivan Hurt, May 03 2022: (Start)
a(n) = A132929(n) - 1.
a(n) = A177336(n) - 2. (End)

A174915 Numbers p such that p, q=p+2 and p+2*q are all primes.

Original entry on oeis.org

3, 5, 11, 41, 59, 101, 179, 191, 269, 311, 431, 521, 599, 821, 881, 1019, 1061, 1151, 1229, 1301, 1451, 1481, 1619, 1721, 1949, 2081, 2111, 2141, 2729, 2999, 3299, 3821, 4001, 4091, 4259, 4421, 4799, 4931, 5009, 5519, 5639, 5849, 6131, 6359, 6689, 6701
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 02 2010

Keywords

Comments

Subsequence of A175914.

Links

Crossrefs

Cf. A001359, A174913, A175914.

Programs

  • Magma
    [p: p in PrimesUpTo(7000) | IsPrime(p+2) and IsPrime(3*p+4)]; // Vincenzo Librandi, Jan 29 2015
  • Mathematica
    lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p1+2*p2],AppendTo[lst,p1]],{n,7!}];lst
Reap[Do[p = Prime[m]; If[PrimeQ[p + 2 ] && PrimeQ[3 p + 4], Sow[p]], {m, 10^3}]][[2, 1]](* Zak Seidov, Oct 14 2012 *)
Transpose[Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==2 && PrimeQ[ #[[1]]+2#[[2]]]&]][[1]] (* Harvey P. Dale, Jan 28 2015 *)
  • PARI
    forprime(p=2,7000,q=p+2;if(isprime(q)&& isprime(p+2*q),print1(p,", ")))

Extensions

Definition and comment corrected by Zak Seidov, Dec 06 2010

A174916 Lesser of twin primes p1 such that p1 + p2^2 - p1^2 is a prime number.

Original entry on oeis.org

3, 5, 11, 17, 29, 71, 101, 281, 311, 419, 461, 521, 599, 617, 641, 659, 809, 827, 857, 881, 1019, 1061, 1277, 1289, 1319, 1607, 1721, 1949, 2027, 2111, 2141, 2309, 2339, 2381, 2591, 2729, 2801, 3329, 3557, 3581, 3767, 3851, 4049, 4127, 4157, 4217, 4229
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 02 2010

Keywords

Comments

3+(5^2-3^2)=3+16=19,..
Let x be the lesser of twin prime pairs. The sequence contains terms such that 5*x+4 is prime. - Harvey P. Dale, Sep 11 2012

Links

Crossrefs

Cf. A001359, A174913, A174915.

Programs

  • Mathematica
    lst={}; Do[p1=Prime[n]; p2=p1+2; If[PrimeQ[p2] && PrimeQ[p1+(p2^2-p1^2)], AppendTo[lst, p1]], {n, 1000}]; lst
With[{ltp=Transpose[Select[Partition[Prime[Range[600]],2,1],#[[2]]- #[[1]]==2&]][[1]]}, Select[ltp,PrimeQ[5#+4]&]] (* Harvey P. Dale, Sep 11 2012 *)

A174917 Lesser of twin primes p1 such that p2+(p2^2-p1^2) is a prime number.

Original entry on oeis.org

5, 11, 29, 41, 107, 137, 149, 197, 239, 347, 431, 461, 569, 599, 659, 809, 821, 1019, 1229, 1289, 1481, 1619, 1787, 1877, 1931, 2027, 2129, 2141, 2309, 2339, 2657, 2687, 2801, 2969, 3119, 3329, 3467, 3557, 3581, 4001, 4019, 4127, 4241, 4421, 4547, 4649
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 02 2010

Keywords

Comments

5+(7^2-5^2)=5+24=29,...

Links

Crossrefs

Cf. A001359, A174913, A174915, A174916

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p2+(p2^2-p1^2)],AppendTo[lst,p1]],{n,4*6!}];lst
Select[Partition[Prime[Range[700]],2,1],#[[2]]-#[[1]]==2&& PrimeQ[ #[[2]]+ #[[2]]^2-#[[1]]^2]&][[All,1]] (* Harvey P. Dale, Dec 18 2021 *)

A174922 Lesser of twin primes p1 such that p1+(p2^2-p1^2) and p2+(p2^2-p1^2) are prime numbers.

Original entry on oeis.org

5, 11, 29, 461, 599, 659, 809, 1019, 1289, 2027, 2141, 2309, 2339, 2801, 3329, 3557, 3581, 4127, 4421, 4547, 5879, 6761, 10091, 10457, 10709, 13829, 15329, 18911, 20231, 21839, 23561, 23909, 26249, 26879, 27581, 27689, 27917, 28109, 30491
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 02 2010

Keywords

Comments

5+(7^2-5^2)=5+24=29; 7+(7^2-5^2)=7+24=31,..

Links

Crossrefs

Cf. A001359, A174913, A174915, A174916, A174917, A174920

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p1+(p2^2-p1^2)]&&PrimeQ[p2+(p2^2-p1^2)],AppendTo[lst,p1]],{n,8!}];lst
prQ[{a_,b_}]:=Module[{c=b^2-a^2},AllTrue[{a+c,b+c},PrimeQ]]; Transpose[ Select[ Select[ Partition[Prime[Range[5000]],2,1],#[[2]]-#[[1]] == 2&], prQ]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 12 2015 *)

A174955 Lesser of twin primes p1 such that p1*p2+-6 are prime numbers.

Original entry on oeis.org

5, 11, 1061, 2111, 3371, 3851, 5867, 9461, 12251, 21491, 22037, 22481, 24917, 26681, 28277, 32141, 42641, 43607, 48731, 56477, 59417, 59627, 67271, 67757, 70487, 77417, 86531, 87221, 91127, 104147, 113621, 115151, 116687, 119291, 121577
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 02 2010

Keywords

Comments

5*7=35+-6 -> primes,..

Crossrefs

Cf. A001359, A174913, A174915, A174916, A174917, A174920, A174922

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p1*p2+6]&&PrimeQ[p1*p2-6],(*Print[p1];*)AppendTo[lst,p1]],{n,8!}];lst

A174957 Lesser of twin primes p1 such that p1*p2-4 and p1*p2-6 are twin prime numbers.

Original entry on oeis.org

5, 11, 1031, 2711, 3851, 4421, 5867, 8837, 10067, 12041, 12251, 12611, 17957, 21491, 21521, 22037, 22481, 23537, 32141, 32411, 42641, 48311, 48731, 49367, 50261, 53231, 60167, 72167, 77417, 80147, 80447, 81047, 87641, 88337, 90527, 95231
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 02 2010

Keywords

Comments

5*7=35; 35-4=31; 35-6=29; 29,31 twin primes

Links

Crossrefs

Cf. A001359, A174913, A174915, A174916, A174917, A174920, A174922, A174955.

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p1*p2-4]&&PrimeQ[p1*p2-6],(*Print[p1];*)AppendTo[lst,p1]],{n,8!}];lst
ltp[{a_,b_}]:=b-a==2&&AllTrue[a*b-{4,6},PrimeQ]; Select[Partition[Prime[ Range[ 10000]],2,1],ltp][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 07 2017 *)
Showing 1-9 of 9 results.

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