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.

A082885 Primes followed by a larger-than-average prime gap.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 73, 83, 89, 113, 131, 139, 151, 157, 167, 173, 181, 199, 211, 233, 241, 251, 257, 263, 271, 283, 293, 317, 331, 337, 353, 359, 367, 373, 383, 389, 401, 409, 421, 449, 467, 479, 491, 509, 523
Offset: 1

Views

Author

Labos Elemer, Apr 17 2003

Keywords

Comments

Previous name was: Primes p(j) such that p(j+1)-p(j) > log(p(j)), where log is the natural logarithm.

Links

Crossrefs

Cf. A082862, A082884, A002386.

Programs

  • Mathematica
    Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s>1, Print[{n, Prime[n], s}]], {n, 1, 1000}]
Transpose[Select[Partition[Prime[Range[100]],2,1],#[[2]]-#[[1]]> Log[#[[1]]]&]][[1]] (* Harvey P. Dale, Mar 15 2015 *)
  • PARI
    p=2;forprime(q=3,1e4,if(q-p>log(p),print1(p", "));p=q) \\ Charles R Greathouse IV, Feb 07 2012

Formula

Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = e (A001113). - Alain Rocchelli, Dec 18 2023

Extensions

New name from Charles R Greathouse IV, Feb 07 2012

A082884 a(n) = smallest prime p for which (r-p)/log(p) < 1/n, where r is the next prime after p.

Original entry on oeis.org

11, 59, 419, 2999, 22037, 162821, 1202627, 8886329, 65660051, 485165279, 3584913989, 26489122349, 195729610331, 1446257064389, 10686474581831, 78962960185097, 583461742527491, 4311231547116551, 31855931757115889
Offset: 1

Views

Author

Labos Elemer, Apr 17 2003

Keywords

Comments

Remark: the quotient can be larger than 1/n at much larger primes, many times. So p does not set record in "standard" sense, only sinks first below 1/n, i.e. smaller than any previous values, but not necessarily smaller than the following ones. See also illustration.

Examples

			a(1)=p(5)=11 and (13-11)/log(11) = 0.8340... < 1/1.

Crossrefs

Cf. A082862, A082885, A082886, A082888-A082891.

Programs

  • Mathematica
    {eq=1, k=1}; Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s<1/k, k=k+1; Print[{k, n, Prime[n], s}]; eq=s], {n, 1, 100000000}]

Formula

a(n)=min{prime(j): A001223(j)/log(prime(j)) < 1/n}, where prime(j)=A000040(j) is the j-th prime.

Extensions

a(11)-a(19) from Donovan Johnson, Sep 09 2008

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

A082886 floor((prime(n+1)-prime(n))/log(prime(n))).

Original entry on oeis.org

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

Views

Author

Labos Elemer, Apr 17 2003

Keywords

Comments

a(n) is unbounded by a theorem of Westzynthius. - Charles R Greathouse IV, Sep 04 2015

Examples

			a(217) = floor((1361-1327)/log(1327)) = floor(4.72834...) = 4.

Links

Crossrefs

Cf. A082862, A082884, A082885, A082888-A082891.

Programs

  • Mathematica
    Table[Floor[(Prime[n+1]-Prime[n])/Log[Prime[n]]//N], {n, 1, 220}]
  • PARI
    a(n,p=prime(n))=(nextprime(p+1)-p)\log(p) \\ Charles R Greathouse IV, Sep 04 2015

Formula

a(n)=floor((prime(n+1)-prime(n))/log(prime(n))).
a(n)=Floor(A001223(n)/log(A000040(n))).
Infinitely often a(n) >> log log n log log log log n/log log log n, see Ford-Green-Konyagin-Maynard-Tao. - Charles R Greathouse IV, Sep 04 2015

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.

A082889 Primes p such that (r-p)/log(p) > 4, where r is the next prime after p.

Original entry on oeis.org

1327, 15683, 16141, 19333, 19609, 20809, 25471, 28229, 31397, 31907, 34061, 34981, 35617, 35677, 36389, 37907, 40289, 40639, 43331, 43801, 44293, 45893, 48679, 58831, 59281, 60539, 69263, 73189, 74959, 79699, 81463, 82073, 85933, 86629
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, A082888, A082890, A082891.

Programs

  • Mathematica
    Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s>4, Print[Prime[n]]], {n, 1, 2000}]
Transpose[Select[Partition[Prime[Range[10000]],2,1],(#[[2]]-#[[1]])/ Log[ #[[1]]]>4&]][[1]] (* Harvey P. Dale, Dec 10 2014 *)

Formula

prime(j) such that (prime(j+1)-prime(j))/log(prime(j)) > 4.

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.

A082892 Floor(q(j)), where q(j) = 2j/log(A000230(j)); log is natural logarithm, 2j-s are prime gaps > 1, A000230(j) is the minimal lesser prime opening the consecutive prime distance equals 2j.

Original entry on oeis.org

1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 7, 8, 8, 8, 7, 8, 8, 8, 9, 8, 8, 9, 8, 8, 8, 9, 10, 9, 9, 10, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 9, 10, 10, 10, 10, 10, 11
Offset: 1

Views

Author

Labos Elemer, Apr 17 2003

Keywords

Comments

For these larger and larger gap-initiating primes, integer part of relevant quotient,q, may exceed 27, all values between 1 and 28 occur. Observation supports conjecture that infsup(q) is infinity.

Crossrefs

Cf. A000230, A082862, A082884-A082891, A002386.

Programs

  • Mathematica
    t=A000230 list; Table[Floor[2*j/Log[Part[t,j]]//N],{j,1,Length[t]}]
Showing 1-9 of 9 results.

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