cp's OEIS Frontend

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

%I A377430 #17 Nov 29 2024 13:50:30
%S A377430 3,4,9,10,13,14,15,22,26,33,39,48,59,60,65,85,88,89,93,104,113,116,
%T A377430 122,142,143,147,148,155,181,188,198,201,209,212,213,224,226,234,235,
%U A377430 244,254,264,265,268,287,288,313,320,328,332,333,341,343,353,361,366
%N A377430 Numbers k such that there is exactly one squarefree number between prime(k)+1 and prime(k+1)-1.
%H A377430 Robert Israel, <a href="/A377430/b377430.txt">Table of n, a(n) for n = 1..10000</a>
%e A377430 Primes 4 and 5 are 7 and 11, and the interval (8,9,10) contains only squarefree 10, so 4 is in the sequence.
%p A377430 R:= NULL: count:= 0: q:= 2:
%p A377430 for k from 1 while count < 100 do
%p A377430   p:= q; q:= nextprime(q);
%p A377430   if nops(select(numtheory:-issqrfree,[$p+1 .. q-1]))=1 then
%p A377430     R:= R,k; count:= count+1;
%p A377430  fi
%p A377430 od:
%p A377430 R; # _Robert Israel_, Nov 29 2024
%t A377430 Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],SquareFreeQ]]==1&]
%o A377430 (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
%Y A377430 For composite instead of squarefree we have A029707.
%Y A377430 These are the positions of 1 in A061398, or 2 in A373198.
%Y A377430 For no squarefree numbers we have A068360.
%Y A377430 For prime-power instead of squarefree we have A377287.
%Y A377430 For at least one squarefree number we have A377431.
%Y A377430 For perfect-power instead of squarefree we have A377434.
%Y A377430 A000040 lists the primes, differences A001223, seconds A036263.
%Y A377430 A002808 lists the composites, complement A008578.
%Y A377430 A005117 lists the squarefree numbers, complement A013929.
%Y A377430 A377038 gives k-differences of squarefree numbers.
%Y A377430 Cf. A007674, A013603, A036378, A046933, A072284, A076259, A077641, A077643, A075526, A078147, A112925, A112926, A120992, A293697.
%K A377430 nonn
%O A377430 1,1
%A A377430 _Gus Wiseman_, Oct 29 2024