cp's OEIS Frontend

A372475 Length of binary expansion (or number of bits) of the n-th squarefree number.

%I A372475 #11 Aug 03 2024 01:52:44
%S A372475 1,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
%T A372475 6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
%U A372475 7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8
%N A372475 Length of binary expansion (or number of bits) of the n-th squarefree number.
%F A372475 a(n) = A070939(A005117(n)).
%F A372475 a(n) = A372472(n) + A372433(n).
%e A372475 The 10th squarefree number is 14, with binary expansion (1,1,1,0), so a(10) = 4.
%t A372475 IntegerLength[Select[Range[1000],SquareFreeQ],2]
%o A372475 (Python)
%o A372475 from math import isqrt
%o A372475 from sympy import mobius
%o A372475 def A372475(n):
%o A372475     def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A372475     m, k = n, f(n)
%o A372475     while m != k:
%o A372475         m, k = k, f(k)
%o A372475     return int(m).bit_length() # _Chai Wah Wu_, Aug 02 2024
%Y A372475 For prime instead of squarefree we have A035100, 1's A014499, 0's A035103.
%Y A372475 Restriction of A070939 to A005117.
%Y A372475 Run-lengths are A077643.
%Y A372475 For weight instead of length we have A372433 (restrict A000120 to A005117).
%Y A372475 For zeros instead of length we have A372472, firsts A372473.
%Y A372475 Positions of first appearances are A372540.
%Y A372475 A030190 gives binary expansion, reversed A030308.
%Y A372475 A048793 lists positions of ones in reversed binary expansion, sum A029931.
%Y A372475 A371571 lists positions of zeros in binary expansion, sum A359359.
%Y A372475 A371572 lists positions of ones in binary expansion, sum A230877.
%Y A372475 A372515 lists positions of zeros in reversed binary expansion, sum A359400.
%Y A372475 Cf. A003714, A023416, A049093, A049094, A059015, A069010, A073642, A145037, A211997, A368494, A372474, A372516.
%K A372475 nonn,base
%O A372475 1,2
%A A372475 _Gus Wiseman_, May 09 2024