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 1-4 of 4 results.

A082891 Smallest prime p such that q = (r-p)/log(p) > n, where r is the next prime after p.

Original entry on oeis.org

2, 7, 1129, 1327, 19609, 31397, 155921, 370261, 1357201, 2010733, 20831323, 20831323, 191912783, 436273009, 3842610773, 10726904659, 25056082087, 25056082087, 25056082087, 1346294310749, 1408695493609, 2614941710599, 13829048559701, 19581334192423, 19581334192423
Offset: 1

Views

Author

Labos Elemer, Apr 17 2003

Keywords

Comments

Is lim superior(q(n)) = +infinity? See A082892.

Examples

			For n = 11 and 12: k = 1319945: p(k+1) = 20831533, p(k) = 20831323, d = p(k+1) - p(k) = 210, log(20831321) = 16.852..., q = 210/16.852... = 12.4615... > 12 and also > 11 for the first time, so a(11) = a(12) = 20831323.

Links

Crossrefs

Cf. A082862, A082884, A082885, A082886, A082888, A082889, A082890, A082892, A111870.

Programs

  • Mathematica
    Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s>11, Print[{n, Prime[n], Prime[n+1], s, Log[Prime[n]]//N}]], {n, 1000000, 100000000}]
  • PARI
    lista(pmax) = {my(n = 1, prv = 2, d, m); print1(2, ", "); forprime(p=3, pmax, d = p-prv; m = floor(d/log(prv)); if(m > n, for(k = 1, m-n, print1(prv, ", ")); n = m); prv=p);} \\ Amiram Eldar, Nov 04 2024

Formula

a(n)= Min{p(x); (p(x+1)-p(x))/log(p(x)) > n}.

Extensions

a(10) corrected and a(13)-a(25) added by Amiram Eldar, Nov 04 2024

A082888 Primes p such that (r-p)/log(p) > 3, where r is the next prime after p.

Original entry on oeis.org

1129, 1327, 1669, 2179, 2477, 2971, 3137, 3271, 4297, 4831, 5119, 5351, 5531, 5591, 5749, 5953, 6491, 6917, 7253, 7759, 7963, 8389, 8467, 8893, 8971, 9551, 9973, 10009, 10399, 10531, 10799, 10909, 11743, 12163, 12853, 13063, 13187, 13933
Offset: 1

Views

Author

Labos Elemer, Apr 17 2003

Keywords

Examples

			If p = 1327 then r = 1361 and (r-p)/log(p) = 34/log(1327) = 4.72834..., so 1327 is in the sequence.

Links

Crossrefs

Cf. A082862, A082884, A082885, A082886, A082889, A082890, A082891.

Programs

  • Mathematica
    Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s>3, Print[Prime[n]]], {n, 1, 2000}]
Transpose[Select[Partition[Prime[Range[2000]],2,1],(Last[#]-First[#])/ Log[ First[ #]]>3&]][[1]] (* Harvey P. Dale, Apr 20 2013 *)

Formula

prime(j) such that (prime(j+1)-prime(j))/log(prime(j)) > 3.

A082890 Primes p such that (r-p)/log(p) > 5, where r is the next prime after p.

Original entry on oeis.org

19609, 25471, 31397, 34061, 35617, 40289, 40639, 43331, 44293, 58831, 79699, 85933, 89689, 102701, 107377, 110359, 124367, 134513, 142993, 149629, 155921, 156157, 162143, 173359, 175141, 186481, 188029, 190409, 203461, 212701, 218287
Offset: 1

Views

Author

Labos Elemer, Apr 17 2003

Keywords

Examples

			If p = 492113 then r = 492227 and (r-p)/log(p) = 114/log(492113) = 8.69799..., so 492113 is in the sequence.

Links

Crossrefs

Cf. A082862, A082884, A082885, A082886, A082888, A082889, A082891.

Programs

  • Mathematica
    Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s>5, Print[Prime[n]]], {n, 1, 50000}]
seq[len_] := Module[{p = 2, s = {}, c = 0}, While[c < len, q = NextPrime[p]; If[q - p > 5*Log[p], AppendTo[s, p]; c++]; p = q]; s]; seq[31] (* Amiram Eldar, Nov 04 2024 *)

Formula

prime(j) such that (prime(j+1)-prime(j))/log(prime(j)) > 5.

A081531 Primes p such that (r-p)/log(p) > 2, where r is the next prime after p.

Original entry on oeis.org

7, 113, 139, 199, 211, 293, 317, 523, 773, 839, 863, 887, 953, 1069, 1129, 1259, 1327, 1381, 1637, 1669, 1759, 1831, 1913, 1933, 1951, 2113, 2161, 2179, 2221, 2251, 2311, 2477, 2503, 2557, 2593, 2803, 2861, 2971, 3089, 3137, 3229, 3271, 3413, 3469, 3739
Offset: 1

Views

Author

Labos Elemer, Apr 17 2003

Keywords

Examples

			If p=1327 then r=1361 and (r-p)/log(p) = 34/log(1327) = 4.72834..., so 1327 is in the sequence.

Links

Crossrefs

Cf. A082862, A082884-A082886, A082889-A082891.

Programs

  • Mathematica
    Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s>2, Print[Prime[n]]], {n, 1, 2000}]
Transpose[Select[Partition[Prime[Range[600]],2,1],(Last[#]-First[#])/ Log[ First[#]]>2&]][[1]] (* Harvey P. Dale, Jun 22 2014 *)

Formula

prime(j) such that (prime(j+1)-prime(j))/log(prime(j)) > 2.
Showing 1-4 of 4 results.

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

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