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A091155 Numbers n such that n - 2^k is squarefree for all 1 <= 2^k < n. 1

%I

%S 2,3,4,7,15,23,39,63,75,87,111,135,147,159,195,219,231,255,267,315,

%T 387,399,411,423,435,447,459,495,519,567,615,663,675,699,711,735,747,

%U 759,771,819,867,915,999,1011,1023,1035,1047,1071,1095,1119,1155,1167,1263

%N Numbers n such that n - 2^k is squarefree for all 1 <= 2^k < n.

%C Erdos conjectures that this sequence is infinite. It appears that n = 3 (mod 12) except for n = 2, 4, 7 and 23.

%D R. K. Guy, Unsolved Problems in Number Theory, A19.

%H T. D. Noe, <a href="/A091155/b091155.txt">Table of n, a(n) for n = 1..10000</a>

%H P. Erd├Ás, <a href="http://www.renyi.hu/~p_erdos/1950-07.pdf">On integers of the form 2^k + p and some related problems</a>, Summa Brasil. Math. 2 (1950), pp. 113-123.

%e 39 is on the list because 38, 37, 35, 31, 23 and 7 are all squarefree.

%t a={}; Do[k=1; While[sf=SquareFreeQ[n-k]; sf&&2k<n, k=2k]; If[sf, AppendTo[a, n]], {n, 2000}]; a

%o (PARI) is(n)=for(k=1,log(n+.5)\log(2),if(!issquarefree(n-2^k),return(0))); 1 \\ _Charles R Greathouse IV_, Apr 13 2014

%Y Cf. A039669 (n such that n-2^k are all primes).

%K nonn

%O 1,1

%A _T. D. Noe_, Dec 23 2003

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Last modified July 20 05:19 EDT 2019. Contains 325168 sequences. (Running on oeis4.)