A354841 -id:A354841 - cp's OEIS frontend

A354842 a(n) is the smallest number k such that A354841(k) = n.

1, 4, 28, 1276, 9430, 463446, 24786396, 1340301868

Offset: 0

Bernard Schott, Jun 24 2022

Equivalently, the smallest integer k such that the number of primes between k and k+log(k), exclusive, is n.

			In the interval ]28; 28+log(28)[ = ]28; 31.332...[, there are two primes 29 and 31 and this is the first such interval with 2 primes, hence a(2) = 28.

Cf. A354840 (similar, but between k and k+log(k)^2), A354841.

seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = Count[Range[n + 1, n + Log[n]], ?PrimeQ] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[5, 10^5] (* _Amiram Eldar, Jun 25 2022 *)

a(3)-a(5) from Alois P. Heinz, Jun 25 2022
a(6) from Amiram Eldar, Jun 25 2022
a(7) from David Consiglio, Jr., Jun 29 2022

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

Alain Rocchelli, Sep 09 2023

nonn

Inspired by A354841.

Cf. A068985, A354841, A275235.

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

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

A366134 Number of primes between prime(n) and prime(n)+2*log(prime(n)), exclusive.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 2, 1, 0, 1, 2, 1, 0, 2, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 0, 0, 3, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 0, 3, 2, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 3, 2, 2, 1, 2, 3, 2, 1, 1, 2, 2, 2, 2

Offset: 1

Alain Rocchelli, Sep 30 2023

nonn

Inspired by A365573.

Cf. A365573, A354841, A092553.

a[n_]:=PrimePi[Prime[n]+2Log[Prime[n]]]-PrimePi[Prime[n]]; Array[a,95] (* Stefano Spezia, Sep 30 2023 *)

a(n) = primepi(prime(n)+2*log(prime(n))) - primepi(prime(n))

Conjecture: Limit_{N->oo} (Ratio_{n=1..N} a(n)=0) = 1/e^2 (A092553).

cp's OEIS Frontend

A354842 a(n) is the smallest number k such that A354841(k) = n.

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

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A366134 Number of primes between prime(n) and prime(n)+2*log(prime(n)), exclusive.

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