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Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.
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%I #19 Dec 13 2024 18:21:54

%S 1,3,10,13,26,33,60,89,104,113,116,142,148,201,209,212,234,265,268,

%T 288,313,320,332,343,353,384,398,408,477,484,498,542,545,551,577,581,

%U 601,625,636,671,719,723,726,745,794,805,815,862,864,884,944,964,995,1054

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

%C 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

%H Amiram Eldar, <a href="/A068361/b068361.txt">Table of n, a(n) for n = 1..10000</a>

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

%F prime(a(n)) = A061351(n). - _Gus Wiseman_, Dec 11 2024

%t Select[Range[100],And@@SquareFreeQ/@Range[Prime[#],Prime[#+1]]&] (* _Gus Wiseman_, Dec 11 2024 *)

%o (PARI) isok(n) = for (k=prime(n)+1, prime(n+1)-1, if (!issquarefree(k), return (0))); 1; \\ _Michel Marcus_, Apr 29 2016

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

%Y Prime index of each (prime) term of A061351.

%Y Positions of zeros in A061399.

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

%Y Positions of zeros in A377784.

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

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

%Y A005117 lists the squarefree numbers, differences A076259.

%Y A006562 finds balanced primes.

%Y A013929 lists the nonsquarefree numbers, differences A078147.

%Y A014574 is the intersection of A006093 and A008864.

%Y A038664 locates the first prime gap of size 2n.

%Y A046933 counts composite numbers between primes.

%Y A061398 counts squarefree numbers between primes, zeros A068360.

%Y A120327 gives the least nonsquarefree number >= n.

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

%K nonn

%O 1,2

%A _Benoit Cloitre_, Feb 28 2002