A097110 -id:A097110 - 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)).

A209308 Denominators of the Akiyama-Tanigawa algorithm applied to 2^(-n), written by antidiagonals.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 4, 4, 8, 8, 1, 4, 8, 4, 16, 2, 2, 1, 8, 32, 32, 1, 2, 4, 4, 16, 32, 64, 8, 8, 16, 16, 64, 64, 128, 128, 1, 8, 16, 8, 32, 64, 128, 32, 256, 2, 2, 8, 16, 64, 64, 128, 64, 512, 512, 1, 2, 4, 8, 32, 64, 128, 16, 128, 512, 1024

Offset: 0

Paul Curtz, Jan 18 2013

1/2^n and successive rows are
1,       1/2,   1/4,   1/8,  1/16,  1/32,   1/64, 1/128, 1/256,...
1/2,     1/2,   3/8,   1/4,  5/32,  3/32,  7/128,  1/32,...       = A000265/A075101, the Oresme numbers n/2^n. Paul Curtz, Jan 18 2013 and May 11 2016
0,       1/4,   3/8,   3/8,  5/16, 15/64, 21/128,...              = (0 before A069834)/new,
-1/4,   -1/4,     0,   1/4, 25/64, 27/64,...
0,      -1/2,  -3/4, -9/16, -5/32,...
1/2,     1/2, -9/16, -13/8,...
0,      17/8, 51/16,...
-17/8, -17/8,...
0
The first column is A198631/(A006519?), essentially the fractional Euler numbers 1, -1/2, 0, 1/4, 0,...  in A060096.
Numerators b(n): 1, 1, 1, 0, 1, 1, -1, 1, 3, 1, ... .
Coll(n+1) - 2*Coll(n) = -1/2, -5/8, -1/2, -11/32, -7/32, -17/128, -5/64, -23/512, ... = -A075677/new, from Collatz problem.
There are three different Bernoulli numbers:
The first Bernoulli numbers are  1, -1/2, 1/6, 0,... = A027641(n)/A027642(n).
The second Bernoulli numbers are 1,  1/2, 1/6, 0,... = A164555(n)/A027642(n). These are the binomial transform of the first one.
The third Bernoulli numbers are  1,   0,  1/6, 0,... = A176327(n)/A027642(n), the half sum. Via A177427(n) and A191567(n), they yield the Balmer series A061037/A061038.
There are three different fractional Euler numbers:
1) The first are  1, -1/2, 0, 1/4, 0, -1/2,... in A060096(n).
Also Akiyama-Tanigawa algorithm for ( 1, 3/2, 7/4, 15/8, 31/16, 63/32,... = A000225(n+1)/A000079(n) ).
2) The second are 1, 1/2, 0, -1/4, 0,  1/2,... , mentioned by Wolfdieter Lang in A198631(n).
3) The third are  0, 1/2, 0, -1/4, 0,  1/2,... , half difference of 2) and 1).
Also Akiyama-Tanigawa algorithm for ( 0, -1/2, -3/4, -7/8, -15/16, -31/32,... =  A000225(n)/A000079(n) ). See A097110(n).

			Triangle begins:
  1,
  2, 2,
  1, 2,  4,
  4, 4,  8,  8,
  1, 4,  8,  4, 16,
  2, 2,  1,  8, 32, 32,
  1, 2,  4,  4, 16, 32,  64,
  8, 8, 16, 16, 64, 64, 128, 128,
  ...

G. C. Greubel, Table of n, a(n) for n = 0..5049
A. F. Horadam, Oresme Numbers, Fibonacci Quarterly, 12, #3, 1974, pp. 267-271.

Cf. Second Bernoulli numbers A164555(n)/A027642(n) via Akiyama-Tanigawa algorithm for 1/(n+1), A272263.

max = 10; t[0, k_] := 1/2^k; t[n_, k_] := t[n, k] = (k + 1)*(t[n - 1, k] - t[n - 1, k + 1]); denoms = Table[t[n, k] // Denominator, {n, 0, max}, {k, 0, max - n}]; Table[denoms[[n - k + 1, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 05 2013 *)

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

A353157 a(n) is the distance from n to the nearest integer whose binary expansion has no common 1-bits with that of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25

Offset: 0

Rémy Sigrist, Apr 27 2022

Equivalently the distance to the nearest integer that can be added without carries in base 2.

			For n = 42 ("101010" in binary):
- 21 ("10101") is the greatest number <= 42 that has no common 1-bits with 42,
- 128 ("1000000") is the least number >= 42 that has no common 1-bits with 42,
- so a(42) = min(42-21, 128-42) = min(21, 86) = 21.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A006257, A020988, A097110, A080079, A108019, A353158.

a(n) = { my (high=2^#binary(n), low=high-1-n); min(n-low, high-n) }

a(n) = min(A006257(n), A080079(n)) for any n > 0.
a(n) = 1 iff n belongs to A097110.
a(n) = n/2 iff n belongs to A020988.
a(n) = n/4 iff n belongs to A108019.
2*a(n) - a(2*n) = 0 or 1.

A163227 Fibonacci-accumulation sequence.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 8, 20, 13, 33, 21, 54, 34, 88, 55, 143, 89, 232, 144, 376, 233, 609, 377, 986, 610, 1596, 987, 2583, 1597, 4180, 2584, 6764, 4181, 10945, 6765, 17710, 10946, 28656, 17711, 46367, 28657, 75024, 46368, 121392, 75025, 196417

Offset: 1

Mark Dols, Jul 23 2009

Accumulation of A000045 and A000071.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,0,0,-1).

Cf. A000071, A000045, A097110.

m:=55; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x-x^2)/(1-2*x^2+x^6))); // Vincenzo Librandi, Dec 12 2016

LinearRecurrence[{0, 2, 0, 0, 0, -1}, {1, 1, 1, 2, 2, 4}, 50] (* or *) CoefficientList[Series[x*(1 + x - x^2)/(1 - 2*x^2 + x^6), {x,1,50}], x] (* G. C. Greubel, Dec 11 2016 *)

Vec(x*(1 + x - x^2)/(1 - 2*x^2 + x^6) + O(x^50)) \\ G. C. Greubel, Dec 11 2016

a(n) = 2*a(n-2) - a(n-6), where a(1,2,3)=1.

G.f.: x*(1 + x - x^2)/(1 - 2*x^2 + x^6). - G. C. Greubel, Dec 11 2016

A303065 Number of numbers < n whose binary representation has the same difference between the numbers of 0's and 1's as n does.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 0, 0, 1, 2, 1, 3, 2, 3, 0, 0, 2, 3, 3, 4, 4, 5, 1, 5, 6, 7, 2, 8, 3, 4, 0, 0, 1, 2, 4, 3, 5, 6, 4, 4, 7, 8, 5, 9, 6, 7, 1, 5, 10, 11, 8, 12, 9, 10, 2, 13, 11, 12, 3, 13, 4, 5, 0, 0, 1, 2, 6, 3, 7, 8, 9, 4, 9, 10, 10, 11, 11, 12, 5, 5, 12, 13

Offset: 0

Alex Ratushnyak, Apr 17 2018

First occurrence of k, k=0,1,2,...: 0, 4, 6, 12, 20, 22, 25, 26, 28, 44, 49, ..., . - Robert G. Wilson v, Feb 08 2018

Ordinal transform of A037861, minus one. - David Radcliffe, May 21 2025

			There are two numbers below 6 with number of 1's in the binary representation minus number of 0's equal to 1, namely 1 and 5, therefore a(6)=2.
There are 3 numbers below 12 with number of 1's in the binary representation minus number of 0's equal to 0, namely 2, 9, 10, therefore a(12)=3.

Alois P. Heinz, Table of n, a(n) for n = 0..32768

Cf. A037861, A097110.

b:= n-> `if`(n=0, 1, add(1-2*i, i=Bits[Split](n))):
p:= proc() -1 end:
a:= proc(n) option remember; local t;
      t:= b(n); p(t):= p(t)+1
    end:
seq(a(n), n=0..82);  # Alois P. Heinz, May 21 2025

d[n_] := DigitCount[n, 2, 1] - DigitCount[n, 2, 0]; f[n_] := Block[{fd = d[n], c = k = 0}, While[k < n, If[d@ k == fd, c++]; k++]; c]; Array[f, 83, 0] (* Robert G. Wilson v, Feb 08 2018 *)

d=[0]*200
for n in range(1024):
    b = bin(n)[2:]
    c0 = b.count('0')
    c1 = len(b) - c0
    diff = c0 - c1
    print(d[100+diff], end=', ')
    d[100+diff] += 1

from collections import Counter
from itertools import count, islice
def a303065_gen():
    counter = Counter()
    for n in count():
        bitstring = format(n, 'b')
        diff = bitstring.count('1') - bitstring.count('0')
        yield counter[diff]
        counter[diff] += 1
a303065_list = list(islice(a303065_gen(), 83)) # David Radcliffe, May 21 2025

a(n) = 0 iff n belongs to A097110. - Rémy Sigrist, May 16 2018

Offset corrected by David Radcliffe, May 21 2025

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

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A209308 Denominators of the Akiyama-Tanigawa algorithm applied to 2^(-n), written by antidiagonals.

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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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A353157 a(n) is the distance from n to the nearest integer whose binary expansion has no common 1-bits with that of n.

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A163227 Fibonacci-accumulation sequence.

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Magma

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A303065 Number of numbers < n whose binary representation has the same difference between the numbers of 0's and 1's as n does.

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