A368084 Squarefree numbers of the form k^2 + k + 1 such that k^2 + k + 2 is also squarefree.

1, 13, 21, 57, 73, 133, 157, 273, 381, 421, 553, 601, 757, 813, 993, 1261, 1333, 1561, 1641, 1893, 1981, 2257, 2353, 2653, 2757, 3081, 3193, 3541, 3661, 4033, 4161, 5113, 5257, 5701, 5853, 6481, 6973, 7141, 7657, 7833, 8373, 8557, 9121, 9313, 9901, 10101, 10713, 10921

Offset

1,2

Comments

Dimitrov (2023) proved that this sequence is infinite.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Stoyan Dimitrov, Square-free pairs n^2 + n + 1, n^2 + n + 2, HAL preprint, hal-03735444, 2023; ResearchGate link.

Formula

a(n) = A002061(A368083(n) + 1).

Example

1 is a term since 1 is squarefree, 1 = 0^2 + 0 + 1, and 0^2 + 0 + 2 = 2 is also squarefree.

Mathematica

Select[Table[n^2 + n + 1, {n, 0, 100}], And @@ SquareFreeQ /@ {#, #+1} &]

Prog

(PARI) lista(kmax) = {my(m); for(k = 0, kmax, m = k^2 + k + 1; if(issquarefree(m) && issquarefree(m + 1), print1(m, ", "))); }

Crossrefs

Intersection of A007674 and A353887.

Cf. A002061, A005117, A368083.

Sequence in context: A053659 A275243 A066630 * A147435 A269309 A203854

Adjacent sequences: A368081 A368082 A368083 * A368085 A368086 A368087

Keyword

nonn,easy

Author

Amiram Eldar, Dec 11 2023

Status

approved