A377430 Numbers k such that there is exactly one squarefree number between prime(k)+1 and prime(k+1)-1.

3, 4, 9, 10, 13, 14, 15, 22, 26, 33, 39, 48, 59, 60, 65, 85, 88, 89, 93, 104, 113, 116, 122, 142, 143, 147, 148, 155, 181, 188, 198, 201, 209, 212, 213, 224, 226, 234, 235, 244, 254, 264, 265, 268, 287, 288, 313, 320, 328, 332, 333, 341, 343, 353, 361, 366

Offset: 1

Author

Gus Wiseman, Oct 29 2024

Keywords

• nonn

Examples

			Primes 4 and 5 are 7 and 11, and the interval (8,9,10) contains only squarefree 10, so 4 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

For composite instead of squarefree we have A029707.

These are the positions of 1 in A061398, or 2 in A373198.

For no squarefree numbers we have A068360.

For prime-power instead of squarefree we have A377287.

For at least one squarefree number we have A377431.

For perfect-power instead of squarefree we have A377434.

A000040 lists the primes, differences A001223, seconds A036263.

A002808 lists the composites, complement A008578.

A005117 lists the squarefree numbers, complement A013929.

A377038 gives k-differences of squarefree numbers.

Cf. A007674, A013603, A036378, A046933, A072284, A076259, A077641, A077643, A075526, A078147, A112925, A112926, A120992, A293697.

Programs

Maple

R:= NULL: count:= 0: q:= 2:
for k from 1 while count < 100 do
  p:= q; q:= nextprime(q);
  if nops(select(numtheory:-issqrfree,[$p+1 .. q-1]))=1 then
    R:= R,k; count:= count+1;
 fi
od:
R; # Robert Israel, Nov 29 2024

Mathematica

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],SquareFreeQ]]==1&]

PARI

is(n,p=prime(n))=my(q=nextprime(p+1),s); for(k=p+1,q-1, if(issquarefree(k) && s++>1, return(0))); s==1 \\ Charles R Greathouse IV, Nov 29 2024