A370600 Numbers m such that 4m + k is squarefree for k = 1..3.

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, 46, 48, 50, 52, 53, 54, 55, 57, 59, 63, 64, 66, 70, 71, 75, 77, 79, 80, 82, 86, 88, 89, 91, 95, 97, 98, 99, 100, 102, 104, 107, 108, 109, 111, 113, 115, 116, 117, 120

Offset: 1

Author

Michael De Vlieger, Apr 10 2024

Keywords

• nonn
• easy
• changed

Comments

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

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.

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

Examples

			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.
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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Plot f(m) at (x,y) = (m mod 361, -floor(m/361)), 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.

Crossrefs

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

Programs

Mathematica

Reap[Do[If[AllTrue[4 n + {1, 2, 3}, SquareFreeQ], Sow[n]], {n, 0, 120}] ][[-1, 1]]
Select[Range[0,150],AllTrue[4#+{1,2,3},SquareFreeQ]&] (* Harvey P. Dale, Aug 19 2025 *)

PARI

is(m) = issquarefree(4*m+1) && issquarefree(4*m+2) && issquarefree(4*m+3); \\ Amiram Eldar, Apr 16 2024

Formula

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