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.

Previous Showing 21-27 of 27 results.

A192320 Numbers k for which there are no prime numbers in the range (k-4*sqrt(sqrt(k)), k].

Original entry on oeis.org

1, 1352, 1353, 1354, 1355, 1356, 1357, 1358, 1359, 1360, 19657, 19658, 19659, 19660, 25522, 31451, 31452, 31453, 31454, 31455, 31456, 31457, 31458, 31459, 31460, 31461, 31462, 31463, 31464, 31465, 31466, 31467, 31468, 34116, 34117, 34118, 34119, 34120, 34121, 34122
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 27 2011

Keywords

Comments

a(69) = 492226 is probably the last term. Any further terms must be greater than 1.5 * 10^18. [Charles R Greathouse IV, Jun 27 2011]

Links

Crossrefs

Subsequence of A192231.
Cf. A000040, A001223, A005669.

Programs

Extensions

a(15) inserted, a(25) and a(34)-a(40) corrected by Charles R Greathouse IV, Jun 27 2011

A085500 Indices of primes where nondecreasing gaps occur.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 15, 16, 18, 21, 23, 24, 30, 62, 66, 99, 154, 189, 217, 1059, 1183, 1532, 1663, 1831, 2225, 2810, 3385, 14357, 29040, 30802, 31545, 40933, 103520, 104071, 118505, 149689, 325852, 733588, 983015, 1094421, 1319945, 2850174, 6957876, 10539432, 10655462
Offset: 1

Views

Author

Farideh Firoozbakht, Aug 15 2003

Keywords

Comments

A005669 is a subsequence of this sequence.

References

  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A8, pp. 31-39.

Links

Crossrefs

Cf. A085237, A005669, A005250, A134266.
See also A214935(n) = A000720(A205827(n)), A241540(n) = A000720(A182514(n)).

Programs

  • Mathematica
    f[n_] := Prime[n+1]-Prime[n]; v1={}; v2={}; Do[If[f[n]>=If[n==1, 1, Last[v2]], v=n; v1=Append[v1, n]; v2=Append[v2, f[v]]; Print[v1]], {n, 105000000}]

Formula

a(n) = A000720(A134266(n)). - M. F. Hasler, Apr 26 2014

Extensions

a(45)-a(47) from Amiram Eldar, Sep 05 2024

A241623 Indices of the primes in A073861 (n-digit primes followed by a maximal gap).

Original entry on oeis.org

2, 24, 154, 1183, 3385, 40933, 325852, 2850174, 23163298, 203615628, 1820471368, 28106444830, 251265078335, 2921439731020, 6822667965940, 49749629143526, 2133658100875638, 20004097201301079
Offset: 1

Views

Author

M. F. Hasler, Apr 26 2014

Keywords

Comments

"Indices" here means the value of the function primepi = A000720, cf. formula.
Subsequence of A005669.

Crossrefs

See also A214935(n) = A000720(A205827(n)), A005669(n) = A000720(A002386(n)).

Formula

a(n) = A000720(A073861(n)).
a(n) = A005669(k) with k such that A002386(k) = A073861(n).

A254033 Number of primes dividing exactly one number in the next largest gap between primes.

Original entry on oeis.org

0, 1, 2, 3, 6, 10, 15, 20, 21, 28, 37, 44, 53, 76, 96, 113, 123, 135, 142, 150, 181, 191, 235, 270, 291, 294, 313, 327, 334, 395, 403, 411, 445, 478, 496, 539, 582, 587, 654, 693, 722, 732, 757, 754, 772, 778, 791, 832, 830, 848, 920, 930, 955, 1004, 1053, 1151, 1240
Offset: 1

Views

Author

Mamuka Jibladze, Jan 23 2015

Keywords

Examples

			The 5th largest prime gap (after 2-3, 3-5, 7-11 and 23-29) occurs between 89 and 97, and there are 6 primes which occur exactly once in this gap, namely 7 (dividing 91), 13 (dividing 91), 19 (dividing 95), 23 (dividing 92), 31 (dividing 93) and 47 (dividing 94), so a(5)=6.

Links

Crossrefs

Sequences related to increasing prime gaps: A005250, A002386, A000101, A005669.

Programs

  • Mathematica
    gp = (* the list of primes in A002386 *); f[n_] := Block[{p = gp[[n]], q = NextPrime[ gp[[n]]]}, r = Range[p + 1, q - 1]; lng = Length@ r; t = Split@ Sort@ Flatten@ Table[ First@# & /@ FactorInteger[ r[[i]]], {i, lng}]; Length@ Select[t, Length@# == 1 &]]; Array[f, 75] (* Robert G. Wilson v, Jan 23 2015 *)

Extensions

a(43)-a(57) from Robert G. Wilson v, Jan 23 2015

A192319 Numbers k such that the half-open interval (k-5*sqrt(sqrt(k)), k] does not contain primes.

Original entry on oeis.org

1, 1358, 1359, 1360, 31464, 31465, 31466, 31467, 31468
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 27 2011

Keywords

Comments

a(9) = 31468 is probably the last term. Any further terms must be greater than 1.5 * 10^18. [Charles R Greathouse IV, Jun 27 2011]

Crossrefs

Subsequence of A192320.
Cf. A000040, A005669.

Programs

  • Maple
    isA192319 := proc(n) phigh := n ; plow := ceil(n-5*root[4](n))-1 ; numtheory[pi](phigh)-numtheory[pi](plow) = 0 ; end proc:
for n from 1 do if isA192319(n) then print(n); end if; end do: # R. J. Mathar, Jul 10 2011
  • PARI
    isA192319(n)=nextprime(floor(n+1-5*n^.25))>n \\ Charles R Greathouse IV, Jun 27 2011

A246796 a(n) = A246795(n) - A246794(n) + 1.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 4, 5, 7, 4, 6, 6, 7, 9, 6, 10, 12, 7, 8, 11, 6, 13, 10, 10, 10, 10, 15, 12, 15, 15, 11, 20, 13, 13, 12, 19, 15, 22, 16, 14, 18, 17, 18, 21, 21, 21, 21, 19, 24, 22, 25, 28, 24, 27, 27, 30, 35, 30, 23, 29, 23, 28, 29, 25, 38, 24, 24, 27, 28, 27, 37, 32, 31, 39
Offset: 2

Views

Author

Farideh Firoozbakht, Oct 29 2014

Keywords

Comments

For a(n) consecutive numbers A005669(n) - k, A246794(n) <= k <= A246795(n), A182134(A005669(n) - k) = k.

Crossrefs

Cf. A005669, A182134, A246794, A246795.

A307325 a(n) is the smallest number k for which prime(k+1) - prime(k) is greater than n.

Original entry on oeis.org

2, 4, 4, 9, 9, 24, 24, 30, 30, 30, 30, 30, 30, 99, 99, 99, 99, 154, 154, 189, 189, 217, 217, 217, 217, 217, 217, 217, 217, 217, 217, 217, 217, 1183, 1183, 1831, 1831, 1831, 1831, 1831, 1831, 1831, 1831, 2225, 2225, 2225, 2225, 2225, 2225, 2225, 2225, 3385, 3385, 3385, 3385
Offset: 1

Views

Author

Marius A. Burtea, Apr 02 2019

Keywords

Comments

For any n there is an infinity of numbers m for which prime(m+1) - prime(m) is greater than n.
It appears that the sequence of lengths of successive runs is equal to A053695. - Marc Bofill Janer, May 21 2019

Examples

			For n = 2, prime(2) - prime(1) = 3 - 2 = 1, prime(3) - prime(2) = 5 - 3 = 2, prime(5) - prime(4) = 11 - 7 = 4, so a(2) = 4.

References

  • Laurențiu Panaitopol, Dinu Șerbănescu, Number theory and combinatorial problems for juniors, Ed.Gil, Zalău, (2003), ch. 1, p.7, pr. 25. (in Romanian).

Crossrefs

Cf. A000040, A001223, A005250, A005669.
Cf. A053695, A144309.

Programs

  • MATLAB
    v=primes(1000000);
for u=1:100; ss=1;
    while and(v(ss+1)-v(ss)<=u,ss
  • Magma
    v:=PrimesUpTo(10000000);
sol:=[];
for u in [1..60] do
   for ss in [1..#v-1] do
    if v[ss+1]-v[ss] gt u then
         sol[u]:=ss;
         break;
     end if;
   end for;
end for;
   sol;
  • PARI
    a(n) = my(k=1); while(prime(k+1) - prime(k) <= n, k++); k; \\ Michel Marcus, Apr 03 2019

Formula

a(2*n) = a(2*n+1) = A144309(n+1) for n>=1. - Georg Fischer, Dec 05 2022
Previous Showing 21-27 of 27 results.

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