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 1-7 of 7 results.

A236119 Primes p with prime(p) - p - 1 and prime(p) - p + 1 both prime.

Original entry on oeis.org

5, 17, 23, 41, 71, 83, 173, 293, 337, 353, 563, 571, 719, 811, 911, 953, 1201, 1483, 1579, 1877, 2081, 2089, 2309, 2579, 2749, 2803, 3329, 3343, 3511, 3691, 3779, 3851, 3881, 3907, 4021, 4049, 4093, 4657, 4813, 5051, 5179, 5333, 5519, 5591, 6053, 6547, 6841, 7151, 7723, 8209
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 19 2014

Keywords

Comments

By the conjecture in A236097, this sequence should have infinitely many terms.

Examples

			a(1) = 5 since neither prime(2) - 2 - 1 = 0 nor prime(3) - 3 - 1 = 1 is prime, but prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are both prime.

Links

Crossrefs

Cf. A000040, A001359, A006512, A014574, A234695, A235925, A236097.

Programs

  • Mathematica
    p[n_]:=PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1100}]
  • PARI
    s=[]; forprime(p=2, 10000, if(isprime(prime(p)-p-1) && isprime(prime(p)-p+1), s=concat(s, p))); s \\ Colin Barker, Jan 19 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}]

A235934 Primes p with f(p), f(f(p)) and f(f(f(p))) all prime, where f(n) = prime(n) - n + 1.

Original entry on oeis.org

2, 3, 23, 311, 1777, 2341, 2861, 3329, 3833, 4051, 8753, 9007, 11587, 13093, 13309, 14551, 16001, 19687, 23143, 26993, 37309, 41981, 44131, 45491, 54623, 56431, 56821, 57991, 60223, 61643, 66413, 66883, 67511, 68767, 69029, 70003, 75743, 76261, 76819, 80021
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 17 2014

Keywords

Comments

By the general conjecture in A235925, this sequence should have infinitely many terms.

Examples

			a(3) = 23 with 23, f(23) = 61, f(61) = 223 and f(223) = 1187 all prime.

Links

Crossrefs

Cf. A000040, A234694, A234695, A235924, A235925.

Programs

  • Mathematica
    f[n_]:=Prime[n]-n+1
p[k_]:=PrimeQ[f[Prime[k]]]&&PrimeQ[f[f[Prime[k]]]]&&PrimeQ[f[f[f[Prime[k]]]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]

A235935 Primes p with f(p), f(f(p)), f(f(f(p))), f(f(f(f(p)))) all prime, where f(n) = prime(n) - n + 1.

Original entry on oeis.org

2, 3, 2861, 8753, 56821, 83449, 162787, 165883, 167197, 186397, 217309, 261721, 275939, 309493, 355571, 382351, 467293, 501187, 539303, 560029, 602839, 640307, 657299, 708959, 879859, 919129, 973813, 1057741, 1085779, 1115899, 1156031, 1302667, 1366297, 1396427, 1516279, 1580461, 1760419, 1829797, 1867249, 1870021
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 17 2014

Keywords

Comments

By the general conjecture in A235925, this sequence should have infinitely many terms.

Examples

			a(3) = 2861 with 2861, f(2861) = 23143, f(23143) = 240769 and f(240769) = 3117791 and f(3117791) =  48951967 all prime.

Links

Crossrefs

Cf. A000040, A234695, A235925, A235934.

Programs

  • Mathematica
    f[n_]:=Prime[n]-n+1
p[k_]:=PrimeQ[f[Prime[k]]]&&PrimeQ[f[f[Prime[k]]]]&&PrimeQ[f[f[f[Prime[k]]]]]&&PrimeQ[f[f[f[f[Prime[k]]]]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,100000}]

A235984 Primes p with f(p), f(f(p)), f(f(f(p))), f(f(f(f(p)))), f(f(f(f(f(p))))) all prime, where f(n) = prime(n) - n + 1.

Original entry on oeis.org

2, 3, 501187, 560029, 2076881, 2836003, 2907011, 8254787, 8822347, 10322189, 11329181, 11354641, 12307693, 14528069, 15801601, 17757427, 19023091, 24995669, 25871971
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 17 2014

Keywords

Comments

By the general conjecture in A235925, this sequence should have infinitely many terms.

Examples

			a(3) = 501187 with 501187, f(501187) = 6886357, f(6886357) = 113948711, f(113948711) = 2224096873, f(2224096873) =  50351471977 and f(50351471977) = 1303792228393 all prime.

Links

Crossrefs

Cf. A000040, A234694, A234695, A235924, A235925, A235934, A235935.

Programs

  • Mathematica
    f[n_]:=Prime[n]-n+1
p[k_]:=PrimeQ[f[Prime[k]]]&&PrimeQ[f[f[Prime[k]]]]&&PrimeQ[f[f[f[Prime[k]]]]]&&PrimeQ[f[f[f[f[Prime[k]]]]]]&&PrimeQ[f[f[f[f[f[Prime[k]]]]]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10^7}]

A236066 Primes p with g(p), g(g(p)), g(g(g(p))), g(g(g(g(p)))), g(g(g(g(g(p))))) all prime, where g(n) = prime(n) - n - 1.

Original entry on oeis.org

5, 98893, 1110709, 4231849, 5319707, 6763349, 7904087, 10823431, 13893109, 15323939, 15544079, 15716713, 17642899, 18978439, 20126237
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 18 2014

Keywords

Comments

Conjecture: For any integer m > 1, there are infinitely many chains p(1) < ... < p(m) of m primes with p(k+1) = prime(p(k)) - p(k) - 1 for all 0 < k < m.
This is similar to the conjecture in A235925.

Examples

			a(1) = 5 since neither g(2) = prime(2) - 2 - 1 = 0 nor g(3) = prime(3) - 3 - 1 = 1 is prime, but 5 = g(5) = g(g(5)) =  g(g(g(5))) = g(g(g(g(5)))) = g(g(g(g(g(5))))) is prime.
a(2) = 98893 with 98893, g(98893) = 1185113, g(1185113) = 17381209, g(17381209) = 304696943, g(304696943) = 6262760333, g(6262760333) = 148561011217 all prime.

Links

Crossrefs

Cf. A000040, A234695, A235925, A235934, A235935, A235984.

Programs

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

A236143 Odd primes p with prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) both prime.

Original entry on oeis.org

7, 11, 31, 67, 179, 193, 197, 281, 347, 349, 563, 599, 757, 1123, 1453, 1543, 1933, 1987, 2083, 2531, 2971, 3037, 3259, 3547, 3583, 3701, 3919, 4027, 4483, 5023, 5581, 5591, 5647, 5981, 6449, 7207, 7297, 7603, 8291, 9049
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 19 2014

Keywords

Comments

By part (i) of the conjecture in A236138, this sequence should have infinitely many terms.

Examples

			a(1) = 7 with prime(6) - 6 = 13 - 6 = 7 and prime(6) - 2*prime(3) = 13 - 2*5 = 3 both prime.

Links

Crossrefs

Cf. A000040, A234695, A235925, A236075, A236119, A236138.

Programs

  • Mathematica
    PQ[n_]:=n>0&&PrimeQ[n]
p[n_]:=PrimeQ[Prime[n-1]-(n-1)]&&PQ[Prime[n-1]-2*Prime[(n-1)/2]]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,2,10^5}]
  • PARI
    s=[]; forprime(p=3, 10000, if(isprime(prime(p-1)-(p-1)) && isprime(prime(p-1)-2*prime((p-1)/2)), s=concat(s, p))); s \\ Colin Barker, Jan 19 2014
Showing 1-7 of 7 results.

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