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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.
7
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
1,2
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.
LINKS
EXAMPLE
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.
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 *)
PROG
(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
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).
Sequence in context: A103841 A003754 A293427 * A087006 A345297 A336533
KEYWORD
nonn
AUTHOR
STATUS
approved