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

1, 65, 137, 209, 217, 281, 353, 433, 641, 713, 785, 793, 857, 937, 1001, 1217, 1289, 1361, 1433, 1505, 1577, 1657, 1793, 1865, 1937, 2081, 2089, 2233, 2305, 2377, 2441, 2513, 2585, 2665, 2729, 2801, 2953, 3017, 3089, 3161, 3241, 3305, 3313, 3457, 3529, 3593

Offset: 1

Author

Franklin T. Adams-Watters, Aug 10 2011

Keywords

• nonn

Comments

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.

From Amiram Eldar, Nov 28 2023: (Start)

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

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)

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Crossrefs

Subsequence of A005117, A007674, A007675 and A017077.

Programs

Mathematica

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 *)

PARI

ap(n)={forstep(k=1,n,8,
if(issquarefree(k)&&issquarefree(k+1)&&issquarefree(k+2)&&
   issquarefree((k+3)\2)&&
   issquarefree(k+4)&&issquarefree(k+5)&&issquarefree(k+6),
  print1(k", ")))}