%I #48 Apr 16 2024 03:06:32

%S 0,1,3,5,7,8,9,10,14,16,17,19,21,23,25,26,27,28,32,34,35,39,41,44,45,

%T 46,48,50,52,53,54,55,57,59,63,64,66,70,71,75,77,79,80,82,86,88,89,91,

%U 95,97,98,99,100,102,104,107,108,109,111,113,115,116,117,120

%N Numbers m such that 4m + k is squarefree for k = 1..3.

%C Numbers m such that A008966(4m+1) + A008966(4m+2) + A008966(4m+3) = 3.

%C The number p^2*m is never squarefree, hence, 4*m is likewise never squarefree. Since 2 is the smallest prime, we have at most 3 consecutive squarefree numbers.

%C The asymptotic density of this sequence is 4 * Product_{p prime} (1 - 3/p^2) = 4 * A206256 = 0.501947... . - _Amiram Eldar_, Apr 16 2024

%H Michael De Vlieger, <a href="/A370600/b370600.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A370600/a370600.png">Plot f(m) at (x,y) = (m mod 361, -floor(m/361))</a>, m = 0..130320, 4X exaggeration, where f(m) = A008966(4m + 1), A008966(4m + 2), A008966(4m + 3), the first term assigned red, second green, and third blue channel. Hence m in this sequence appear white, while those in A258332 appear black.

%F a(n) = (A007675(n)-1)/4.

%e For m = 0, all of {4(0)+1, 4(0)+2, 4(0)+3} = {1, 2, 3} are squarefree and composite; these are all squarefree semiprimes. Hence, 0 is in the sequence.

%e For m = 2, {4(2)+1, 4(2)+2, 4(2)+3} = {9, 10, 11} only the latter 2 numbers are squarefree. Therefore, 2 is not in the sequence.

%t Reap[Do[If[AllTrue[4 n + {1, 2, 3}, SquareFreeQ], Sow[n]], {n, 0, 120}] ][[-1, 1]]

%o (PARI) is(m) = issquarefree(4*m+1) && issquarefree(4*m+2) && issquarefree(4*m+3); \\ _Amiram Eldar_, Apr 16 2024

%Y Cf. A005117, A007675, A008966, A206256, A258332.

%K nonn,easy

%O 1,3

%A _Michael De Vlieger_, Apr 10 2024