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A368084
Squarefree numbers of the form k^2 + k + 1 such that k^2 + k + 2 is also squarefree.
2
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
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.
Sequence in context: A053659 A275243 A066630 * A147435 A269309 A203854
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Dec 11 2023
STATUS
approved