A293430 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.

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

Offset: 1

Author

Antti Karttunen and Michael De Vlieger, Oct 10 2017

Keywords

• nonn

Comments

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

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

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Marked terms in the binary tree illustration of A293230.

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

Positions of nonzero terms in A293233.

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

Programs

Mathematica

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 *)

PARI

is_persistently_squarefree(n,base) = { while(n>1, if(!issquarefree(n),return(0)); n \= base); (1); };
isA293430(n) = is_persistently_squarefree(n,2);
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