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A372433 Binary weight (number of ones in binary expansion) of the n-th squarefree number.

%I A372433 #19 Feb 14 2025 19:39:56
%S A372433 1,1,2,2,2,3,2,3,3,3,4,2,3,3,3,4,3,4,4,5,2,2,3,3,3,4,3,3,4,4,5,4,4,5,
%T A372433 4,4,5,5,5,2,2,3,3,3,4,3,3,4,4,5,3,4,4,4,5,4,5,5,5,6,3,4,4,5,4,4,5,5,
%U A372433 5,6,4,4,5,5,6,5,6,7,2,2,3,3,3,3,3,4,4
%N A372433 Binary weight (number of ones in binary expansion) of the n-th squarefree number.
%H A372433 Harvey P. Dale, <a href="/A372433/b372433.txt">Table of n, a(n) for n = 1..1000</a>
%H A372433 MathOverflow, <a href="http://mathoverflow.net/questions/22629">Are there primes of every Hamming weight?</a>
%H A372433 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hamming_weight">Hamming weight</a>.
%F A372433 a(n) = A000120(A005117(n)).
%F A372433 a(n) + A372472(n) = A372475(n) = A070939(A005117(n)).
%t A372433 DigitCount[Select[Range[100],SquareFreeQ],2,1]
%t A372433 Total[IntegerDigits[#,2]]&/@Select[Range[200],SquareFreeQ] (* _Harvey P. Dale_, Feb 14 2025 *)
%o A372433 (Python)
%o A372433 from math import isqrt
%o A372433 from sympy import mobius
%o A372433 def A372433(n):
%o A372433     def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A372433     m, k = n, f(n)
%o A372433     while m != k:
%o A372433         m, k = k, f(k)
%o A372433     return int(m).bit_count() # _Chai Wah Wu_, Aug 02 2024
%Y A372433 Restriction of A000120 to A005117.
%Y A372433 For prime instead of squarefree we have A014499, zeros A035103.
%Y A372433 Counting zeros instead of ones gives A372472, cf. A023416, A372473.
%Y A372433 For binary length instead of weight we have A372475.
%Y A372433 A003714 lists numbers with no successive binary indices.
%Y A372433 A030190 gives binary expansion, reversed A030308.
%Y A372433 A048793 lists positions of ones in reversed binary expansion, sum A029931.
%Y A372433 A145037 counts ones minus zeros in binary expansion, cf. A031443, A031444, A031448, A097110.
%Y A372433 A371571 lists positions of zeros in binary expansion, sum A359359.
%Y A372433 A371572 lists positions of ones in binary expansion, sum A230877.
%Y A372433 A372515 lists positions of zeros in reversed binary expansion, sum A359400.
%Y A372433 A372516 counts ones minus zeros in binary expansion of primes, cf. A177718, A177796, A372538, A372539.
%Y A372433 Cf. A039004, A049093, A049094, A059015, A069010, A070939, A073642, A211997, A368494, A372474.
%K A372433 nonn,base
%O A372433 1,3
%A A372433 _Gus Wiseman_, May 04 2024