A293430 - OEIS

%I #27 Oct 18 2017 20:38:54

%S 1,2,3,5,6,7,10,11,13,14,15,21,22,23,26,29,30,31,42,43,46,47,53,58,59,

%T 61,62,85,86,87,93,94,95,106,107,118,119,122,123,170,173,174,186,187,

%U 190,191,213,214,215,237,238,239,246,247,341,346,347,349,373,374,381,382,383,426,427,429,430,431,474,478,479

%N Persistently squarefree numbers for base-2 shifting: Numbers n such that all terms in finite set [n, floor(n/2), floor(n/4), floor(n/8), ..., 1] are squarefree.

%C Question: Is this sequence infinite? (My guess: yes). This is equivalent to questions asked in A293230. See also comments at A293441 and A293517.

%C For any odd n that is present, 2n is also present.

%H Antti Karttunen, <a href="/A293430/b293430.txt">Table of n, a(n) for n = 1..10000</a>

%e For 479 we see that 479 is prime (thus squarefree, in A005117), [479/2] = 239 is also a prime, [239/2] = 119 = 7*17 (a squarefree composite), [119/2] = 59 (a prime), [59/2] = 29 (a prime), [29/2] = 14 = 2*7 (a squarefree composite), [14/2] = 7 (a prime), [7/2] = 3 (a prime), [3/2] = 1 (the end of halving process 1 is also squarefree), thus all the values obtained by repeated halving were squarefree and 479 is a member of this sequence. Here [ ] stands for taking floor.

%t With[{s = Fold[Append[#1, MoebiusMu[#2] #1[[Floor[#2/2]]]] &, {1}, Range[2, 480]]}, Flatten@ Position[s, _?(# != 0 &)]] (* _Michael De Vlieger_, Oct 10 2017 *)

%o (PARI)

%o is_persistently_squarefree(n,base) = { while(n>1, if(!issquarefree(n),return(0)); n \= base); (1); };

%o isA293430(n) = is_persistently_squarefree(n,2);

%o n=0; k=1; while(k <= 10000, n=n+1; if(isA293430(n),write("b293430.txt", k, " ", n);k=k+1)); \\ _Antti Karttunen_, Oct 11 2017

%Y Marked terms in the binary tree illustration of A293230.

%Y Subsequence of A293427 (thus also of A003754 and of A005117).

%Y Positions of nonzero terms in A293233.

%Y Cf. A293441, A293517, A293523 (for floor(n/3^k) analog), A293437 (for a subsequence).

%K nonn

%O 1,2

%A _Antti Karttunen_ and _Michael De Vlieger_, Oct 10 2017