A357315 Numbers m such that for all k < m, at least one of m*k - 1 and m*k + 1 is squarefree.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 30, 32, 36, 42, 44, 48, 50, 52, 54, 56, 62, 64, 66, 70, 72, 78, 84, 90, 96, 126, 132, 140, 144, 150, 156, 168, 180, 198, 210, 216, 228, 240, 246, 264, 270, 360, 378, 390, 414, 420, 450, 510, 546, 630, 780, 840, 1230, 1470, 1680, 5250

Offset

1,2

Comments

Conjecture: this sequence is finite.

a(70) > 2*10^7, if it exists. - Giovanni Resta, Oct 20 2022

Links

Sidney Cadot, Table of n, a(n) for n = 1..69

Example

11 is not in this sequence because 11*5-1=54, 11*5+1=56 are both squareful numbers and 11*9-1=98, 11*9+1=100 are both squareful numbers.

Mathematica

q[n_] := AllTrue[Range[n - 1]*n, SquareFreeQ[# - 1] || SquareFreeQ[# + 1] &]; Select[Range[2000], q] (* Amiram Eldar, Oct 20 2022 *)

Prog

(Magma) [n: n in [1..2000] | #[k: k in [1..n-1] | not IsSquarefree(n*k-1) and not IsSquarefree(n*k+1)] eq 0];
(PARI) isok(m) = for (k=1, m-1, if (!issquarefree(m*k - 1) && !issquarefree(m*k + 1), return(0)); ); return(1); \\ Michel Marcus, Oct 20 2022

Crossrefs

Cf. A005117, A013929, A257771, A281192.

Sequence in context: A056757 A349898 A225737 * A079333 A063938 A002473

Adjacent sequences: A357312 A357313 A357314 * A357316 A357317 A357318

Keyword

nonn

Author

Juri-Stepan Gerasimov, Oct 17 2022

Extensions

Name edited by Thomas Scheuerle, Oct 20 2022

a(69) from Amiram Eldar, Oct 20 2022

Status

approved