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Numbers k such that k and k+3 are consecutive squarefree numbers.
2

%I #12 Nov 12 2021 04:36:34

%S 7,23,26,43,62,74,79,115,119,134,146,151,167,170,174,187,206,223,259,

%T 274,278,287,295,314,323,331,359,362,367,374,386,403,439,458,494,506,

%U 511,523,527,530,538,566,574,583,619,623,635,638,655,674,691,710,727,734

%N Numbers k such that k and k+3 are consecutive squarefree numbers.

%C The asymptotic density of this sequence is 0.0716601371276261... (Mossinghoff et al., 2021).

%H Amiram Eldar, <a href="/A349230/b349230.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, <a href="https://doi.org/10.1090/mcom/3581">The distribution of k-free numbers</a>, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; <a href="https://arxiv.org/abs/1912.04972">arXiv preprint</a>, arXiv:1912.04972 [math.NT], 2019-2020.

%e 7 is a term since 7 and 7 + 3 = 10 = 2*5 are squarefree, and 7 + 1 = 8 = 2^3 and 7 + 2 = 9 = 3^2 are not.

%t Select[Range[750], Boole[SquareFreeQ /@ (# + {0, 1, 2, 3})] == {1, 0, 0, 1} &]

%o (PARI) isok(k) = issquarefree(k) && !issquarefree(k+1) && !issquarefree(k+2) && issquarefree(k+3); \\ _Michel Marcus_, Nov 11 2021

%Y Cf. A005117, A007674, A020754, A280892, A349231.

%K nonn

%O 1,1

%A _Amiram Eldar_, Nov 11 2021