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Numbers k that are the start of a sequence of 7 maximally-squarefree numbers.
3

%I #21 Nov 28 2023 10:57:56

%S 1,65,137,209,217,281,353,433,641,713,785,793,857,937,1001,1217,1289,

%T 1361,1433,1505,1577,1657,1793,1865,1937,2081,2089,2233,2305,2377,

%U 2441,2513,2585,2665,2729,2801,2953,3017,3089,3161,3241,3305,3313,3457,3529,3593

%N Numbers k that are the start of a sequence of 7 maximally-squarefree numbers.

%C k, k+1, k+2, k+4, k+5, and k+6 are squarefree; k+3 is divisible by 4 but no higher power of 2 and no other prime squared.

%C From _Amiram Eldar_, Nov 28 2023: (Start)

%C All the terms are of the form 8*k + 1.

%C The numbers of terms not exceeding 10^k for k = 1, 2, ... , are 1, 2, 14, 140, 1384, 13774, 137784, 1378053, 13779491, 137794128, 1377940943, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0137794... . (End)

%H Amiram Eldar, <a href="/A194002/b194002.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)

%t sfQ[n_]:=Module[{c4=FactorInteger[n[[4]]],r=Drop[n,{4}]},First[c4] == {2,2} && Max[Transpose[Rest[c4]][[2]]]==1&&And@@SquareFreeQ/@r]; Join[{1}, Transpose[ Select[Partition[Range[2,3600],7,1],sfQ]][[1]]] (* _Harvey P. Dale_, Nov 22 2011 *)

%o (PARI) ap(n)={forstep(k=1,n,8,

%o if(issquarefree(k)&&issquarefree(k+1)&&issquarefree(k+2)&&

%o issquarefree((k+3)\2)&&

%o issquarefree(k+4)&&issquarefree(k+5)&&issquarefree(k+6),

%o print1(k", ")))}

%Y Subsequence of A005117, A007674, A007675 and A017077.

%K nonn

%O 1,2

%A _Franklin T. Adams-Watters_, Aug 10 2011