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.

A038623 Smallest prime p such that p/pi(p)>=n.

Original entry on oeis.org

2, 2, 37, 127, 347, 1087, 3109, 8419, 24317, 64553, 175211, 480881, 1304707, 3523901, 9558533, 25874843, 70115473, 189961529, 514272533, 1394193607, 3779851091, 10246935679, 27788566133, 75370121191, 204475052401, 554805820711, 1505578023841, 4086199302113
Offset: 1

Views

Author

Jud McCranie

Keywords

Examples

			pi(37)=12 and a(3)=37 is the smallest prime >= 3*12.

Links

Crossrefs

Cf. A038624-A038627.
Essentially the same as A062743,A038607.
a(n) = prime(A038624(n)).

Programs

  • Mathematica
    Prime[Join[{k = 1}, Table[While[Prime[k]/k < n, k++]; k, {n, 2, 18}]]] (* Jayanta Basu, Jul 10 2013 *)
  • PARI
    k=n=1; forprime(p=2,, while(p/k>=n, print1(p", "); n++); k++) \\ Charles R Greathouse IV, Oct 15 2016

Formula

a(n) = exp(n + 1 + o(1)). - Charles R Greathouse IV, Oct 15 2016

Extensions

Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar
a(24)-a(28) from David W. Wilson, Apr 25 2017
a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A062742 Index j of prime p(j) such that floor(p(j)/j) = n is first satisfied.

Original entry on oeis.org

2, 1, 12, 31, 69, 181, 443, 1052, 2701, 6455, 15928, 40073, 100362, 251707, 637235, 1617175, 4124437, 10553415, 27066974, 69709680, 179992909, 465769803, 1208198526, 3140421716, 8179002096, 21338685407, 55762149030, 145935689361
Offset: 1

Views

Author

Labos Elemer, Jul 12 2001

Keywords

Examples

			The q(j)=p(j)/j quotient when the value 14 first appears: {j=251706, p(j)=3523841, q(j)=13.9998291} {251707, 3523901, 14.0000119} {251708, 3523903, 13.9999642} {251709, 3523921, 13.9999801} {251710, 3523957, 14.0000675} {251711, 3523963, 14.0000357}

Links

Crossrefs

Essentially the same as A038624.
Cf. A038606. - R. J. Mathar, Jan 30 2009

Programs

  • PARI
    {a062742(m)=local(n,j); for(n=1,m,j=1; while(floor(prime(j)/j)!=n,j++); print1(j,","))} a062742(10^7)

Formula

a(n) = Min_{j| floor(p(j)/j) = n}. Note that neither p(j)/j nor floor(p(j)/j) is monotonic.
a(n) = pi(A062743(n)).
a(n) = A038606(n) = A038624(n) for n >= 3. - Jaroslav Krizek, Dec 13 2009

Extensions

More terms from Jason Earls, May 15 2002
a(17)-a(28) from Farideh Firoozbakht and Robert G. Wilson v, Sep 13 2005
a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A062357 a(n) = n*p(n+1)-(n+1)*p(n) = n*d(n)-p(n), where p(n) is the n-th prime and d(n) is the n-th prime-difference, A001223(n).

Original entry on oeis.org

-1, 1, 1, 9, -1, 11, -3, 13, 31, -9, 35, 11, -15, 13, 43, 43, -25, 47, 9, -31, 53, 9, 55, 103, 3, -49, 5, -51, 7, 307, -3, 61, -71, 201, -79, 65, 65, -11, 67, 67, -97, 239, -105, -17, -107, 353, 353, -31, -129, -29, 73, -135, 289, 73, 73, 73, -155, 77, -41, -161, 327, 575, -55, -183, -53, 607, 71, 343, -209, -69, 73, 217
Offset: 1

Views

Author

Labos Elemer, Jul 13 2001

Keywords

Comments

A sequence based on the solution of the equation: 1+(1+n)*prime(n)/x-n*prime(n+1)/x=0 for x. This is an irrational rotation-like sequence: the sequence is similar to a Beatty sequence. - Roger L. Bagula, Jun 06 2002

Examples

			n = 10: a(10) = 10*31-11*29 = 310-319 = -9;
n = 54: a(54) = 54*257-55*251 = 13878-13805 = 73;
n = 55: a(55) = 55*263-56*257 = 14465-14392 = 73; consecutive terms are often equal to each other.

Links

Crossrefs

Cf. A000040, A001223, A062742, A062743.

Programs

  • Magma
    [n*NthPrime(n + 1) - (n + 1)*NthPrime(n): n in [1..75]]; // Vincenzo Librandi, Jun 29 2018
  • Maple
    seq(n*ithprime(n+1)-(n+1)*ithprime(n),n=1..80); # Muniru A Asiru, Jun 29 2018
  • Mathematica
    Table[(Prime[w+1]-Prime[w])*w-Prime[w], {w, 1, 1024}]
  • PARI
    a(n)={n*prime(n + 1) - (n + 1)*prime(n)} \\ Harry J. Smith, Aug 06 2009

Formula

a(n) = n*A000040(n+1) - (n+1)*A000040(n) = n*A001223(n) - A000040(n).

A364635 a(n) is the largest prime p such that p/PrimePi(p) < n.

Original entry on oeis.org

7, 31, 113, 359, 1129, 3089, 8467, 24281, 64717, 175141, 481447, 1304713, 3524621, 9560081, 25874773, 70119967, 189969349, 514282961, 1394199299, 3779856617, 10246936393, 27788573801, 75370126379, 204475055189, 554805820519, 1505578026059, 4086199303001, 11091501632977
Offset: 2

Views

Author

Jon E. Schoenfield, Sep 09 2023

Keywords

Comments

Sequence begins at a(2) because there exists no prime p such that p/PrimePi(p) < 1.

Examples

			a(4) = 113 because 113/PrimePi(113) = 113/30 = 3.766... but p/PrimePi(p) >= 4 for all primes > 113.

Crossrefs

Cf. A000720, A038625, A062743, A102281.

Formula

a(n) = prime(A102281(n)). - Michel Marcus, Sep 10 2023
Showing 1-4 of 4 results.

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

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