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 11 results. Next

A236458 Primes p with p + 2 and prime(p) + 2 both prime.

Original entry on oeis.org

3, 5, 17, 41, 1949, 2309, 2711, 2789, 2801, 3299, 3329, 3359, 3917, 4157, 4217, 4259, 4637, 5009, 5021, 5231, 6449, 7757, 8087, 8219, 8627, 9419, 9929, 10007, 10937, 11777, 12071, 14321, 15647, 15971, 16061, 16901, 18131, 18251, 18287, 18539
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 26 2014

Keywords

Comments

According to the conjecture in A236470, the sequence should have infinitely many terms. This is stronger than the twin prime conjecture.
See A236457 and A236467 for similar sequences.

Examples

			a(1) = 3 since 3 + 2 = 5 and prime(3) + 2 = 7 are both prime, but 2 + 2 = 4 is composite.

Links

Crossrefs

Cf. A000040, A001359, A006512, A236119, A236456, A236457, A236467, A236470.

Programs

  • Mathematica
    p[n_]:=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]
  • PARI
    s=[]; forprime(p=2, 20000, if(isprime(p+2) && isprime(prime(p)+2), s=concat(s, p))); s \\ Colin Barker, Jan 26 2014

A236456 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(n-k)/4 - 1, q = p + 2 and r = prime(q) + 2 are all prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 3, 3, 2, 4, 2, 2, 4, 5, 4, 2, 3, 1, 3, 2, 3, 4, 3, 4, 5, 0, 2, 2, 3, 2, 4, 2, 4, 3, 2, 2, 1, 2, 5, 2, 3, 1, 4, 2, 2, 4, 1, 4, 1, 5, 4, 2, 2, 1, 2, 1, 5, 3, 3, 1, 2, 2, 4, 1, 3, 4, 2, 2, 1, 0, 2, 4, 2, 1, 3, 1, 4, 3, 5, 3, 2, 1, 3, 2, 3, 2, 0
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 26 2014

Keywords

Comments

Conjecture: a(n) > 0 for all n > 357.
This is much stronger than the twin prime conjecture. Actually it implies that there are infinitely many primes p such that {p, p + 2} and {prime(p+2), prime(p+2) + 2} are both twin prime pairs. See A236457 for such primes p.

Examples

			a(18) = 1 since 18 = 3 + 15 with phi(3) + phi(15)/4 - 1  = 3,  3 + 2 = 5 and prime(5) + 2 = 13 all prime.
a(50) = 1 since 50 = 16 + 34 with phi(16) + phi(34)/4 - 1 = 11, 11 + 2 = 13 and prime(13) + 2 = 43 all prime.
a(929) = 1 since 929 = 441 + 488 with phi(441) + phi(488)/4 - 1 = 252 + 60 - 1 = 311, 311 + 2 = 313 and prime(313) + 2 = 2083 all prime.

Links

Crossrefs

Cf. A000010, A000040, A001359, A006512, A236097, A236457, A236458.

Programs

  • Mathematica
      p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n+2]+2]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/4-1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236464 Primes p with prime(p) + 2 and prime(p) + 6 both prime.

Original entry on oeis.org

3, 5, 7, 13, 43, 89, 313, 613, 643, 743, 1171, 1279, 1627, 1823, 1867, 1999, 2311, 2393, 2683, 2753, 2789, 3571, 4441, 4561, 5039, 5231, 5647, 5953, 6067, 6317, 6899, 8039, 8087, 8753, 8923, 9337, 9787, 9931, 10259, 10667
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 26 2014

Keywords

Comments

According to the conjecture in A236472, this sequence contains infinitely many terms, i.e., there are infinitely many prime triples of the form {prime(p), prime(p) + 2, prime(p) + 6} with p prime.
See A236462 for a similar sequence.

Examples

			a(1) = 3 since 3, prime(3) + 2 = 7 and prime(3) + 6 = 11 are all prime, but prime(2) + 6 = 9 is composite.

Links

Crossrefs

Cf. A000040, A022004, A236457, A236458, A236462, A236472.

Programs

  • Mathematica
    p[n_]:=p[n]=PrimeQ[Prime[n]+2]&&PrimeQ[Prime[n]+6]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]

A236467 Primes p with p + 2 and prime(p) - 2 both prime.

Original entry on oeis.org

3, 11, 29, 149, 179, 191, 269, 347, 431, 461, 617, 659, 1031, 1619, 1931, 3467, 3527, 4799, 6569, 6689, 7349, 7877, 9011, 9767, 11117, 12611, 13691, 13901, 14549, 16067, 16139, 16451, 16631, 17489, 17681, 18911, 20981, 22367, 23909, 24179
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 26 2014

Keywords

Comments

According to the conjecture in A236468, this sequence should have infinitely many terms.
See A236457 and A236458 for similar sequences.

Examples

			a(1) = 3 since 3, 3 + 2 = 5 and prime(3) - 2 = 3 are all prime, but 2 + 2 = 4 is composite.

Links

Crossrefs

Cf. A000040, A001359, A006512, A236119, A236457, A236458, A236462, A236464, A236468.

Programs

  • Mathematica
    p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]-2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]
Select[Prime[Range[3000]],AllTrue[{#+2,Prime[#]-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 11 2020 *)

A236462 Primes p with prime(p) + 4 and prime(p) + 6 both prime.

Original entry on oeis.org

19, 59, 151, 181, 211, 229, 389, 571, 877, 983, 1039, 1259, 1549, 3023, 3121, 3191, 3259, 3517, 3719, 4099, 4261, 4463, 5237, 6947, 7529, 7591, 7927, 7933, 8317, 8389, 8971, 9403, 9619, 10163, 10939, 11131, 11717, 11743, 11839, 12301
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 26 2014

Keywords

Comments

According to the conjecture in A236460, this sequence should have infinitely many terms.
See A236464 for a similar sequence.

Examples

			a(1) = 19 with 19, prime(19) + 4 = 71 and prime(19) + 6 = 73 all prime.

Links

Crossrefs

Cf. A000040, A022005, A236456, A236457, A236458, A236460, A236464.

Programs

  • Mathematica
    p[n_]:=p[n]=PrimeQ[Prime[n]+4]&&PrimeQ[Prime[n]+6]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]
Select[Prime[Range[1500]],AllTrue[Prime[#]+{4,6},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 21 2018 *)
  • PARI
    s=[]; forprime(p=2, 12500, if(isprime(prime(p)+4) && isprime(prime(p)+6), s=concat(s, p))); s \\ Colin Barker, Jan 26 2014

A236481 Primes p with p + 2, prime(p) + 2 and prime(prime(p)) + 2 all prime.

Original entry on oeis.org

3, 1949, 4217, 8219, 9929, 22091, 23537, 28097, 38711, 41609, 50051, 60899, 68111, 72227, 74159, 79631, 115151, 122399, 127679, 150959, 155537, 266687, 267611, 270551, 271499, 284741, 306347, 428297, 433661, 444287
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 26 2014

Keywords

Comments

Conjecture: For any positive integer m, there are infinitely many chains p(1) < p(2) < ... < p(m) of m primes with p(k) + 2 prime for all k = 1,...,m such that p(k + 1) = prime(p(k)) for every 0 < k < m.

Examples

			a(1) = 3 since 3, 3 + 2 = 5, prime(3) + 2 = 7 and prime(prime(3)) + 2 = prime(5) + 2 = 13 are all prime, but 2 + 2 = 4 is composite.

Links

Crossrefs

Cf. A000010, A000040, A001359, A006512, A235925, A236066, A236457, A236458, A236468, A236470, A236480, A236482, A236484.

Programs

  • Mathematica
    p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10^6}]

A236460 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 - 1, prime(p) + 4 and prime(p) + 6 are all prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 2, 2, 1, 1, 2, 0, 4, 1, 2, 1, 5, 2, 1, 3, 1, 1, 3, 2, 6, 3, 0, 2, 5, 5, 6, 3, 4, 5, 3, 4, 4, 4, 6, 3, 2, 6, 2, 3, 2, 10, 2, 3, 1, 6, 1, 4, 0, 2, 3, 4, 2, 4, 0, 4, 0, 3, 2, 3, 0, 4, 0, 1, 1, 4
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 26 2014

Keywords

Comments

Conjecture: a(n) > 0 for all n > 211.
This implies that there are infinitely many primes p with {prime(p), prime(p) + 4, prime(p) + 6} a prime triple. See A236462 for such primes p.

Examples

			a(30) = 1 since 30 = 13 + 17 with phi(13) + phi(17)/2 - 1 = 19, prime(19) + 4 = 67 + 4 = 71 and prime(19) + 6 = 73 all prime.
a(831) = 1 since 831 = 66 + 765 with phi(66) + phi(765)/2 - 1 = 20 + 192 - 1 = 211, prime(211) + 4 = 1297 + 4 = 1301 and prime(211) + 6 = 1303 all prime.

Links

Crossrefs

Cf. A000010, A000040, A001359, A006512, A022005, A236456, A236457, A236458, A236462, A236464.

Programs

  • Mathematica
    p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+4]&&PrimeQ[Prime[n]+6]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2-1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

A236482 Primes p with p + 2, prime(p) + 2, prime(prime(p)) + 2 and prime(prime(prime(p))) + 2 all prime.

Original entry on oeis.org

41609, 1119047, 1928621, 2348579, 2371709, 3406727, 4098569, 4204817, 4438997, 5561819, 6161159, 6293297, 8236439, 8736701, 8890667, 8951387, 9231329, 9390077, 10492457, 10619897, 11255729, 11514719, 11769479, 11920661, 12316697
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 26 2014

Keywords

Comments

According to the general conjecture in A236481, this sequence should have infinitely many terms.

Examples

			a(1) = 41609 with 41609, 41609 + 2 = 41611, prime(41609) + 2 = 500909 + 2 = 500911, prime(500909) + 2 = 7382957 + 2 = 7382959 and prime(7382957) + 2 = 130090109 + 2 = 130090111 all prime.

Links

Crossrefs

Cf. A000040, A001359, A006512, A236457, A236458, A236467, A236480, A236481, A236484.

Programs

  • Mathematica
    p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]&&PrimeQ[Prime[Prime[Prime[n]]]+2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10^6}]

A236568 Primes p with prime(p + 2) + 2 prime.

Original entry on oeis.org

3, 5, 11, 31, 41, 43, 47, 67, 79, 107, 199, 223, 251, 263, 311, 313, 461, 467, 499, 577, 599, 641, 727, 743, 797, 911, 919, 929, 1163, 1187, 1277, 1303, 1429, 1433, 1447, 1613, 1619, 1621, 1637, 1783, 1789, 1823, 1831, 1867, 1879, 1997, 2029, 2039, 2089, 2309
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 28 2014

Keywords

Comments

This sequence is interesting because of the conjecture in A236566.
Note that A236457 is a subsequence of the sequence.

Examples

			a(1) = 3 since prime(3 + 2) + 2 = 11 + 2 = 13 is prime, but prime (2 + 2) + 2 = 9 is not.

Links

Crossrefs

Cf. A000040, A001359, A006512, A236457, A236566.

Programs

  • Mathematica
    p[n_]:=PrimeQ[Prime[n+2]+2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]
Select[Prime[Range[400]],PrimeQ[Prime[#+2]+2]&] (* Harvey P. Dale, Feb 13 2024 *)

A236480 a(n) = |{0 < k < n-2: p = 2*phi(k) + phi(n-k)/2 + 1, prime(p) + 2 and prime(prime(p)) + 2 are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 2, 1, 3, 2, 2, 0, 2, 3, 1, 2, 1, 3, 3, 2, 2, 1, 1, 1, 3, 0, 2, 3, 2, 1, 3, 0, 2, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 26 2014

Keywords

Comments

Conjecture: a(n) > 0 for every n = 640, 641, ....
We have verified this for n up to 75000.
The conjecture implies that there are infinitely many primes p with prime(p) + 2 and prime(prime(p)) + 2 both prime.

Examples

			a(8) = 1 since 2*phi(3) + phi(5)/2 + 1 = 7, prime(7) + 2 = 17 + 2 = 19 and prime(prime(7)) + 2 = prime(17) + 2 = 61 are all prime.
a(667) = 1 since 2*phi(193) + phi(667-193)/2 + 1 = 384 + 78 + 1 = 463, prime(463) + 2 = 3299 + 2 = 3301 and prime(prime(463)) + 2 = prime(3299) + 2 = 30559 are all prime.

Links

Crossrefs

Cf. A000010, A000040, A001359, A006512, A236456, A236457, A236458, A236468, A236470, A236481.

Programs

  • Mathematica
    p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]
f[n_,k_]:=2*EulerPhi[k]+EulerPhi[n-k]/2+1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]
Showing 1-10 of 11 results. Next

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

Content is available under The OEIS End-User License Agreement.