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.

A354840 a(n) is the smallest number k such that A275235(k) = n.

Original entry on oeis.org

1, 4, 9, 16, 35, 57, 93, 222, 427, 819, 1257, 1270, 1276, 2651, 5806, 13673, 19366, 19372, 27723, 108857, 113036, 113038, 115748, 524856, 560074, 1006146, 1219767, 1652728, 2704892, 2704894, 8756936, 21401949, 21401979, 40268383, 40268435, 40268437, 167540089, 167540101
Offset: 0

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Author

Bernard Schott, Jun 08 2022

Keywords

Comments

Equivalently, the smallest integer k such that the number of primes between k and k+log(k)^2, exclusive, is n.
Up to a(37) = 167540101, the last known term, this sequence is monotonic.

Examples

			In the interval ]9; 9+log(9)^2[ = ]9; 13.827...[, there are two primes 11 and 13 and this is the first such interval with two primes, hence a(2) = 9.

References

  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A2, Primes connected with factorials, p. 11.

Crossrefs

Cf. A275235, A354842 (similar, but between k and k+log(k)).

Programs

  • Mathematica
    f[n_] := Count[Range[n + 1, n + Log[n]^2], ?PrimeQ]; seq[len, max_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < max, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[15, 10^4] (* Amiram Eldar, Jun 08 2022 *)

Extensions

More terms from Amiram Eldar, Jun 08 2022

A354841 Number of primes between n and n+log(n), exclusive.

Original entry on oeis.org

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

Views

Author

Bernard Schott, Jun 08 2022

Keywords

Comments

Inspired by A275235.

Crossrefs

Cf. A275235 (similar, but with n+log(n)^2), A354842.

Programs

A365573 Number of primes between prime(n) and prime(n)+log(prime(n)), exclusive.

Original entry on oeis.org

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

Views

Author

Alain Rocchelli, Sep 09 2023

Keywords

Comments

Inspired by A354841.

Crossrefs

Cf. A068985, A354841, A275235.

Programs

  • PARI
    a(n) = primepi(prime(n)+log(prime(n))) - primepi(prime(n))

Formula

Conjecture: Limit_{N->oo} (Ratio_{n=1..N} a(n)=0) = 1/e (A068985).
Showing 1-3 of 3 results.

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

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