A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

1, 3, 10, 13, 26, 33, 60, 89, 104, 113, 116, 142, 148, 201, 209, 212, 234, 265, 268, 288, 313, 320, 332, 343, 353, 384, 398, 408, 477, 484, 498, 542, 545, 551, 577, 581, 601, 625, 636, 671, 719, 723, 726, 745, 794, 805, 815, 862, 864, 884, 944, 964, 995, 1054

Offset: 1

Author

Benoit Cloitre, Feb 28 2002

Keywords

• nonn

Comments

Also numbers k such that all numbers from prime(k) to prime(k+1) are squarefree. All such primes are twins, so this is a subset of A029707. The other twin primes are A061368. - Gus Wiseman, Dec 11 2024

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

A subset of A029707 (lesser index of twin primes).

Prime index of each (prime) term of A061351.

Positions of zeros in A061399.

For perfect power instead of squarefree we have A377436, zeros of A377432.

Positions of zeros in A377784.

The rest of the twin primes are at A378620, indices of A061368.

A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).

A005117 lists the squarefree numbers, differences A076259.

A006562 finds balanced primes.

A013929 lists the nonsquarefree numbers, differences A078147.

A014574 is the intersection of A006093 and A008864.

A038664 locates the first prime gap of size 2n.

A046933 counts composite numbers between primes.

A061398 counts squarefree numbers between primes, zeros A068360.

A120327 gives the least nonsquarefree number >= n.

Cf. A000720, A007674, A013928, A057627, A070321, A071403, A072284, A073247, A112925, A122535, A155752, A224363, A240473, A251092.

Programs

Mathematica

Select[Range[100],And@@SquareFreeQ/@Range[Prime[#],Prime[#+1]]&] (* Gus Wiseman, Dec 11 2024 *)

PARI

isok(n) = for (k=prime(n)+1, prime(n+1)-1, if (!issquarefree(k), return (0))); 1; \\ Michel Marcus, Apr 29 2016

Formula

n such that A061398(n) = prime(n+1)-prime(n)-1.

prime(a(n)) = A061351(n). - Gus Wiseman, Dec 11 2024