A372515 -id:A372515 - cp's OEIS frontend

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

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, 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, 5, 6, 4, 4, 5, 5, 6, 5, 6, 7, 2, 2, 3, 3, 3, 3, 3, 4, 4

Offset: 1

Gus Wiseman, May 04 2024

Harvey P. Dale, Table of n, a(n) for n = 1..1000
MathOverflow, Are there primes of every Hamming weight?
Wikipedia, Hamming weight.

Restriction of A000120 to A005117.
For prime instead of squarefree we have A014499, zeros A035103.
Counting zeros instead of ones gives A372472, cf. A023416, A372473.
For binary length instead of weight we have A372475.
A003714 lists numbers with no successive binary indices.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A145037 counts ones minus zeros in binary expansion, cf. A031443, A031444, A031448, A097110.
A371571 lists positions of zeros in binary expansion, sum A359359.
A371572 lists positions of ones in binary expansion, sum A230877.
A372515 lists positions of zeros in reversed binary expansion, sum A359400.
A372516 counts ones minus zeros in binary expansion of primes, cf. A177718, A177796, A372538, A372539.
Cf. A039004, A049093, A049094, A059015, A069010, A070939, A073642, A211997, A368494, A372474.

DigitCount[Select[Range[100],SquareFreeQ],2,1]
Total[IntegerDigits[#,2]]&/@Select[Range[200],SquareFreeQ] (* Harvey P. Dale, Feb 14 2025 *)

from math import isqrt
from sympy import mobius
def A372433(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m).bit_count() # Chai Wah Wu, Aug 02 2024

a(n) = A000120(A005117(n)).

a(n) + A372472(n) = A372475(n) = A070939(A005117(n)).

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

Original entry on oeis.org

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, 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, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Gus Wiseman, May 09 2024

			The 10th squarefree number is 14, with binary expansion (1,1,1,0), so a(10) = 4.

For prime instead of squarefree we have A035100, 1's A014499, 0's A035103.
Restriction of A070939 to A005117.
Run-lengths are A077643.
For weight instead of length we have A372433 (restrict A000120 to A005117).
For zeros instead of length we have A372472, firsts A372473.
Positions of first appearances are A372540.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A371571 lists positions of zeros in binary expansion, sum A359359.
A371572 lists positions of ones in binary expansion, sum A230877.
A372515 lists positions of zeros in reversed binary expansion, sum A359400.
Cf. A003714, A023416, A049093, A049094, A059015, A069010, A073642, A145037, A211997, A368494, A372474, A372516.

IntegerLength[Select[Range[1000],SquareFreeQ],2]

from math import isqrt
from sympy import mobius
def A372475(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m).bit_length() # Chai Wah Wu, Aug 02 2024

a(n) = A070939(A005117(n)).

a(n) = A372472(n) + A372433(n).

A372473 Least k such that the k-th squarefree number has exactly n zeros in its binary expansion.

Original entry on oeis.org

1, 2, 7, 12, 21, 40, 79, 158, 315, 1247, 1246, 2492, 4983, 9963, 19921, 39845, 79689, 159361, 318726, 637462, 1274919, 2549835, 5099651, 10199302, 20398665, 40797328, 81594627, 163189198, 326378285, 652756723, 1305513584, 2611027095, 5222054082, 10444108052

Offset: 0

Gus Wiseman, May 09 2024

Note that the data is not strictly increasing.

			The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:
     1:              1 ~ {1}
     2:             10 ~ {2}
    10:           1010 ~ {2,4}
    17:          10001 ~ {1,5}
    33:         100001 ~ {1,6}
    65:        1000001 ~ {1,7}
   129:       10000001 ~ {1,8}
   257:      100000001 ~ {1,9}
   514:     1000000010 ~ {2,10}
  2051:   100000000011 ~ {1,2,12}
  2049:   100000000001 ~ {1,12}
  4097:  1000000000001 ~ {1,13}
  8193: 10000000000001 ~ {1,14}

Chai Wah Wu, Table of n, a(n) for n = 0..57

Positions of first appearances in A372472.
For prime instead of squarefree we have A372474, A035103, A372517, A014499.
Counting bits (length) gives A372540, firsts of A372475, runs A077643.
Counting 1's (weight) instead of 0's gives A372541, firsts of A372433.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A070939 gives length of binary expansion (number of bits).
A371571 lists positions of zeros in binary expansion, sum A359359.
A371572 lists positions of ones in binary expansion, sum A230877.
A372515 lists positions of zeros in reversed binary expansion, sum A359400.
Cf. A035100, A039004, A049093, A049094, A059015, A069010, A145037, A211997.

nn=10000;
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
dcs=DigitCount[Select[Range[nn],SquareFreeQ],2,0];
Table[Position[dcs,i][[1,1]],{i,0,spnm[dcs]}]

from math import isqrt
from itertools import count
from sympy import factorint, mobius
from sympy.utilities.iterables import multiset_permutations
def A372473(n):
    if n==0: return 1
    for l in count(n):
        m = 1<Chai Wah Wu, May 10 2024

a(23)-a(33) from Chai Wah Wu, May 10 2024

A372472 Number of zeros in the binary expansion of the n-th squarefree number.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 0, 3, 2, 2, 2, 1, 2, 1, 1, 0, 4, 4, 3, 3, 3, 2, 3, 3, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 5, 5, 4, 4, 4, 3, 4, 4, 3, 3, 2, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1, 4, 3, 3, 2, 3, 3, 2, 2, 2, 1, 3, 3, 2, 2, 1, 2, 1, 0, 6, 6, 5, 5, 5, 5, 5, 4, 4

Offset: 1

Gus Wiseman, May 09 2024

			The 12th squarefree number is 17, with binary expansion (1,0,0,0,1), so a(12) = 3.

Positions of first appearances are A372473.
Restriction of A023416 to A005117.
For prime instead of squarefree we have A035103, ones A014499, bits A035100.
Counting 1's instead of 0's (so restrict A000120 to A005117) gives A372433.
For binary length we have A372475, run-lengths A077643.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A371571 lists positions of zeros in binary expansion, sum A359359.
A371572 lists positions of ones in binary expansion, sum A230877.
A372515 lists positions of zeros in reversed binary expansion, sum A359400.
Cf. A003714, A039004, A049093, A049094, A059015, A069010, A070939, A073642, A145037, A211997, A368494, A372474, A372516.

A372583 := proc(n)
    A023416(A005117(n)) ;
end proc:
seq(A372583(n),n=1..200) ; # R. J. Mathar, May 20 2024

DigitCount[Select[Range[100],SquareFreeQ],2,0]

a(n) = A023416(A005117(n)).

a(n) + A372433(n) = A070939(A005117(n)) = A372475(n).

A372541 Least k such that the k-th squarefree number has exactly n ones in its binary expansion.

Original entry on oeis.org

1, 3, 6, 11, 20, 60, 78, 157, 314, 624, 1245, 3736, 4982, 9962, 19920, 39844, 79688, 239046, 318725, 956194, 1912371, 2549834, 5099650, 15298984, 20398664, 40797327, 81594626, 163189197, 326378284, 979135127, 1305513583, 2611027094, 5222054081, 10444108051

Offset: 0

Gus Wiseman, May 09 2024

			The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:
       1:                   1 ~ {1}
       3:                  11 ~ {1,2}
       7:                 111 ~ {1,2,3}
      15:                1111 ~ {1,2,3,4}
      31:               11111 ~ {1,2,3,4,5}
      95:             1011111 ~ {1,2,3,4,5,7}
     127:             1111111 ~ {1,2,3,4,5,6,7}
     255:            11111111 ~ {1,2,3,4,5,6,7,8}
     511:           111111111 ~ {1,2,3,4,5,6,7,8,9}
    1023:          1111111111 ~ {1,2,3,4,5,6,7,8,9,10}
    2047:         11111111111 ~ {1,2,3,4,5,6,7,8,9,10,11}
    6143:       1011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13}
    8191:       1111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13}
   16383:      11111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14}
   32767:     111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
   65535:    1111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}
  131071:   11111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}

Chai Wah Wu, Table of n, a(n) for n = 0..56
Wikipedia, Hamming weight.

Positions of firsts appearances in A372433.
Counting zeros instead of ones gives A372473, firsts in A372472.
For prime instead of squarefree we have A372517, firsts of A014499.
Counting bits (length) gives A372540, firsts of A372475, runs A077643.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A145037, A097110 count ones minus zeros, for primes A372516, A177796.
A371571 lists positions of zeros in binary expansion, sum A359359.
A371572 lists positions of ones in binary expansion, sum A230877.
A372515 lists positions of zeros in reversed binary expansion, sum A359400.
Cf. A023416, A049093, A049094, A069010, A070939, A071403, A211997, A280296, A372474.

nn=10000;
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
dcs=DigitCount[Select[Range[nn],SquareFreeQ],2,1];
Table[Position[dcs,i][[1,1]],{i,spnm[dcs-1]}]

from math import isqrt
from itertools import count
from sympy import factorint, mobius
from sympy.utilities.iterables import multiset_permutations
def A372541(n):
    if n==0: return 1
    for l in count(n):
        m = 1<Chai Wah Wu, May 10 2024

a(23)-a(33) from Chai Wah Wu, May 10 2024

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

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

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A372473 Least k such that the k-th squarefree number has exactly n zeros in its binary expansion.

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A372472 Number of zeros in the binary expansion of the n-th squarefree number.

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A372541 Least k such that the k-th squarefree number has exactly n ones in its binary expansion.

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