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 1-3 of 3 results.

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}]
Showing 1-3 of 3 results.

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