cp's OEIS Frontend

A194002 Numbers k that are the start of a sequence of 7 maximally-squarefree numbers.

%I A194002 #21 Nov 28 2023 10:57:56
%S A194002 1,65,137,209,217,281,353,433,641,713,785,793,857,937,1001,1217,1289,
%T A194002 1361,1433,1505,1577,1657,1793,1865,1937,2081,2089,2233,2305,2377,
%U A194002 2441,2513,2585,2665,2729,2801,2953,3017,3089,3161,3241,3305,3313,3457,3529,3593
%N A194002 Numbers k that are the start of a sequence of 7 maximally-squarefree numbers.
%C A194002 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 A194002 From _Amiram Eldar_, Nov 28 2023: (Start)
%C A194002 All the terms are of the form 8*k + 1.
%C A194002 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 A194002 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 A194002 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 A194002 (PARI) ap(n)={forstep(k=1,n,8,
%o A194002 if(issquarefree(k)&&issquarefree(k+1)&&issquarefree(k+2)&&
%o A194002    issquarefree((k+3)\2)&&
%o A194002    issquarefree(k+4)&&issquarefree(k+5)&&issquarefree(k+6),
%o A194002   print1(k", ")))}
%Y A194002 Subsequence of A005117, A007674, A007675 and A017077.
%K A194002 nonn
%O A194002 1,2
%A A194002 _Franklin T. Adams-Watters_, Aug 10 2011