A080791 -id:A080791 - cp's OEIS frontend

A372540 Least k such that the k-th squarefree number has binary expansion of length n. Index of the smallest squarefree number >= 2^n.

1, 2, 4, 7, 12, 21, 40, 79, 158, 315, 625, 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 10 2024

			The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:
       1:                  1 ~ {1}
       2:                 10 ~ {2}
       5:                101 ~ {1,3}
      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}
    1027:        10000000011 ~ {1,2,11}
    2049:       100000000001 ~ {1,12}
    4097:      1000000000001 ~ {1,13}
    8193:     10000000000001 ~ {1,14}
   16385:    100000000000001 ~ {1,15}
   32770:   1000000000000010 ~ {2,16}
   65537:  10000000000000001 ~ {1,17}
  131073: 100000000000000001 ~ {1,18}

Chai Wah Wu, Table of n, a(n) for n = 0..73

Counting zeros instead of length gives A372473, firsts of A372472.
For prime instead of squarefree we have:
- zeros A372474, firsts of A035103
- ones A372517, firsts of A014499
- bits A372684, firsts of A035100
Positions of first appearances in A372475, run-lengths A077643.
For weight instead of length we have A372541, firsts of A372433.
Indices of the squarefree numbers listed by A372683.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers.
A030190 gives binary expansion, reversed A030308.
A070939 counts bits, binary length, or length of binary expansion.
Cf. A029931, A048793, A049093, A049094, A059015, A069010, A143658, A211997, A230877.

nn=1000;
ssnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[Max@@#]&];
dcs=IntegerLength[Select[Range[nn],SquareFreeQ],2];
Table[Position[dcs,i][[1,1]],{i,ssnm[dcs]}]

from itertools import count
from math import isqrt
from sympy import mobius, factorint
def A372540(n): return next(sum(mobius(a)*(k//a**2) for a in range(1, isqrt(k)+1)) for k in count(1<Chai Wah Wu, May 12 2024

A005117(a(n)) = A372683(n).

a(n) = A143658(n)+1 for n > 1. - Chai Wah Wu, Aug 26 2024

a(24)-a(34) from Chai Wah Wu, May 12 2024

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

A372474 Least k such that the k-th prime number has exactly n zeros in its binary expansion.

Original entry on oeis.org

2, 1, 8, 7, 19, 32, 99, 55, 174, 310, 565, 1029, 1902, 3513, 6544, 6543, 23001, 43395, 82029, 155612, 295957, 564164, 1077901, 3957811, 3965052, 7605342, 14630844, 28194383, 54400029, 105097568, 393615809, 393615807, 762939128, 1480206930, 2874398838, 5586502349

Offset: 0

Gus Wiseman, May 11 2024

			The prime numbers A000040(a(n)) together with their binary expansions and binary indices begin:
         3:                          11 ~ {1,2}
         2:                          10 ~ {2}
        19:                       10011 ~ {1,2,5}
        17:                       10001 ~ {1,5}
        67:                     1000011 ~ {1,2,7}
       131:                    10000011 ~ {1,2,8}
       523:                  1000001011 ~ {1,2,4,10}
       257:                   100000001 ~ {1,9}
      1033:                 10000001001 ~ {1,4,11}
      2053:                100000000101 ~ {1,3,12}
      4099:               1000000000011 ~ {1,2,13}
      8209:              10000000010001 ~ {1,5,14}
     16417:             100000000100001 ~ {1,6,15}
     32771:            1000000000000011 ~ {1,2,16}
     65539:           10000000000000011 ~ {1,2,17}
     65537:           10000000000000001 ~ {1,17}
    262147:         1000000000000000011 ~ {1,2,19}
    524353:        10000000000001000001 ~ {1,7,20}
   1048609:       100000000000000100001 ~ {1,6,21}
   2097169:      1000000000000000010001 ~ {1,5,22}
   4194433:     10000000000000010000001 ~ {1,8,23}
   8388617:    100000000000000000001001 ~ {1,4,24}
  16777729:   1000000000000001000000001 ~ {1,10,25}
  67108913: 100000000000000000000110001 ~ {1,5,6,27}
  67239937: 100000000100000000000000001 ~ {1,18,27}

Positions of first appearances in A035103.
For squarefree instead of prime we have A372473, firsts of A372472.
Counting ones (weight) gives A372517, firsts of A014499.
Counting squarefree bits gives A372540, firsts of A372475, runs A077643.
Counting squarefree ones gives A372541, firsts of A372433.
Counting bits (length) gives A372684, firsts of A035100.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
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).
Cf. A059015, A066195, A069010, A073642, A145037, A211997, A230877, A359359, A359400, A371571, A372516.

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

from itertools import count
from sympy import isprime, primepi
from sympy.utilities.iterables import multiset_permutations
def A372474(n):
    for l in count(n):
        m = 1<Chai Wah Wu, May 13 2024

a(n) = A000720(A066195(n)). - Robert Israel, May 13 2024

a(22)-a(35) from and offset corrected by Chai Wah Wu, May 13 2024

A059448 The parity of the number of zero digits when n is written in binary.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1

Offset: 1

Henry Bottomley, Feb 02 2001

Old name was: "If A_k are the terms from 2^(k-1) through to 2^k-1, then A_(k+1) is B_k A_k where B_k is A_k with 0's and 1's swapped, starting from a(1)=0; also parity of number of zero digits when n is written in binary. a(0) not given as it could be 1 or 0 depending on the definition or formula used." - Michel Dekking, Sep 11 2020
The sequence (when prefixed by 0) is overlap-free [Allouche and Shallit].
From Vladimir Shevelev, May 23 2017: (Start)
Theorem: The sequence is cubefree.
Here we show only that the sequence contains no three consecutive equal terms. Indeed, using the recursions below, we have
a(4*n)=a(n), a(4*n+1)=1-a(n), a(4*n+2)=1-a(n), a(4*n+3)=a(n), n >= 1, and our statement easily follows. In general, the Theorem could be proved either directly (cf. A269027) or using the remark below from Jeffrey Shallit and the well-known fact [first proved not later than 1912 by Axel Thue (private communication from Jean-Paul Allouche)] that the Thue-Morse sequence is cubefree.
Note that, by the formulas modulo 4, the sequence is constructed over four terms {a(4*n),a(4*n+1),a(4*n+2),a(4*n+3)} which, starting with a(4), are either {0,1,1,0} or {1,0,0,1}, the first elements of which form {a(n)}. (End)

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 26, Problem 23.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jeffrey Shallit, Arseny M. Shur, and Stefan Zorcic, New constructions for 3-free and 3+-free binary morphisms, arXiv:2310.15064 [math.CO], 2023. Mentions this sequence.
Index entries for sequences related to binary expansion of n
Index entries for characteristic functions

Characteristic function of A059009.
Cf. A298952 (complement), A242179 (values +-1).
Cf. A023416 (A080791), A054429, A010059, A010060, A106400.

a059448 = (`mod` 2) . a023416  -- Reinhard Zumkeller, Mar 01 2012

s1:=[];
for n from 1 to 200 do
t1:=convert(n,base,2); t2:=subs(1=NULL,t1); s1:=[op(s1),nops(t2) mod 2]; od:
s1;

Table[Boole[OddQ[Count[IntegerDigits[n, 2], 0]]], {n, 1, 105}] (* Jean-François Alcover, Apr 05 2013 *)

a(n)=(#binary(n)-hammingweight(n))%2;
vector(99,n,a(n)) /* Joerg Arndt, Sep 11 2020 */

def A059448(n): return (n.bit_length()^n.bit_count())&1 # Chai Wah Wu, Jul 26 2023

a(2n) = 1 - a(n); a(2n+1) = a(n) = 1 - a(2n). If 2^k <= n < 2^(k+1) then a(n) = 1 - a(n-2^(k-1)). a(n) = A023416(n) mod 2 = A059009(n) - 2n = 2n + 1 - A059010(n) = |A010060(n) - A030300(n-1)|.
Let b(1)=1 and b(n) = b(n-ceiling(n/2)) - b(n-floor(n/2)); then for n >= 1, a(n) = (1/2)*(1-b(2n+1)). - Benoit Cloitre, Apr 26 2005
Alternatively, if x is the sequence, then x = 010 mu^2(x), where mu is the Thue-Morse morphism sending 0 to 01 and 1 to 10. - Jeffrey Shallit, Jun 06 2016
a(n) = A010059(A054429(n)) = (1+A008836(A163511(n)))/2. - Antti Karttunen, May 30 2017
Alternatively, if x is the sequence, then x = 0 tau(x), where tau is the "twisted" Thue-Morse morphism sending 0 to 10 and 1 to 01. Note that tau^2 = mu^2, giving x = 010 mu^2(x). - Michel Dekking, Sep 30 2020

Name changed by Michel Dekking, Sep 11 2020

A346633 Sum of even-indexed parts (even bisection) of the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 01 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Composition number 741 in standard order is (2,1,1,3,2,1), so a(741) = 1 + 3 + 1 = 5.

Including odd-indexed parts gives A029837.
Subtracting from the odd version gives A124754.
Positions of zeros are A131577.
The odd-indexed version is A209281(n+1).
The version for prime indices is A346698 (reverse: A346700).
A000120 and A080791 count binary digits 1 and 0, with difference A145037.
A011782 counts compositions.
A056239 adds up prime indices, row sums of A112798.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345197 counts compositions by sum, length, and alternating sum.
Cf. A000070, A000097, A008549, A025047, A088218, A344606, A344653, A346697.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total[Last/@Partition[Append[stc[n],0],2]],{n,0,100}]

a(n) = (A029837(n) - A124754(n))/2.
a(n) = A029837(n) - A209281(n + 1).
a(n) = A124754(n) + A209281(n + 1).

A372517 Least k such that the k-th prime number has exactly n ones in its binary expansion.

Original entry on oeis.org

1, 2, 4, 9, 11, 64, 31, 76, 167, 309, 502, 801, 1028, 7281, 6363, 12079, 12251, 43237, 43390, 146605, 291640, 1046198, 951351, 2063216, 3957778, 11134645, 14198321, 28186247, 54387475, 249939829, 105097565, 393248783, 751545789, 1391572698, 2182112798, 8242984130

Offset: 1

Gus Wiseman, May 12 2024

In other words, the a(n)-th prime is the least with binary weight n. The sorted version is A372686.

			The primes A000040(a(n)) together with their binary expansions and binary indices begin:
        2:                     10 ~ {2}
        3:                     11 ~ {1,2}
        7:                    111 ~ {1,2,3}
       23:                  10111 ~ {1,2,3,5}
       31:                  11111 ~ {1,2,3,4,5}
      311:              100110111 ~ {1,2,3,5,6,9}
      127:                1111111 ~ {1,2,3,4,5,6,7}
      383:              101111111 ~ {1,2,3,4,5,6,7,9}
      991:             1111011111 ~ {1,2,3,4,5,7,8,9,10}
     2039:            11111110111 ~ {1,2,3,5,6,7,8,9,10,11}
     3583:           110111111111 ~ {1,2,3,4,5,6,7,8,9,11,12}
     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}
    73727:      10001111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,17}
    63487:       1111011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13,14,15,16}

Positions firsts of first appearances in A014499.
Taking primes gives A061712.
Counting zeros (weight) gives A372474, firsts of A035103.
For binary length we have A372684 (take primes A104080), firsts of A035100.
The sorted version is A372686, taking primes A372685.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A029837 gives greatest binary index, least A001511.
A030190 gives binary expansion, reversed A030308.
A048793 lists binary indices, reverse A272020, sum A029931.
A372471 lists binary indices of primes.
Cf. A000040, A066195, A069010, A070939, A071814, A211997, A372429, A372516.

spsm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[Max@@#]&];
j=DigitCount[#,2,1]&/@Select[Range[1000],PrimeQ];
Table[Position[j,k][[1,1]],{k,spsm[j]}]

a(n) = my(k=1, p=2); while(hammingweight(p) !=n, p = nextprime(p+1); k++); k; \\ Michel Marcus, May 13 2024

from itertools import count
from sympy import isprime, primepi
from sympy.utilities.iterables import multiset_permutations
def A372517(n):
    for l in count(n-1):
        m = 1<Chai Wah Wu, May 13 2024

A000040(a(n)) = A061712(n).

a(32)-a(36) from Pontus von Brömssen, May 13 2024

A257806 a(n) = A257808(n) - A257807(n).

Original entry on oeis.org

0, -1, 0, 1, 0, 1, 2, 1, 2, 1, 2, 3, 2, 3, 4, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 7, 6, 7, 8, 7, 6, 7, 8, 9, 10, 11, 12, 11, 12, 13, 12, 11, 10, 9, 10, 9, 10, 11, 10, 11, 12, 13, 12, 11, 12, 13, 12, 13, 12, 13, 14, 13, 12, 11, 10, 9, 10, 11, 12, 11, 10, 9, 10, 11, 12, 13, 14, 15, 14, 15, 16, 15, 16, 15, 14

Offset: 0

Antti Karttunen, May 12 2015

Alternative description: Start with a(0) = 0, and then to obtain each a(n), look at each successive term in the infinite trunk of inverted binary beanstalk, from A233271(1) onward, subtracting one from a(n-1) if A233271(n) is odd, and adding one to a(n-1) if A233271(n) is even.
In other words, starting from zero, iterate the map x -> {x + 1 + number of nonleading zeros in the binary representation of x}, and note each time whether the result is odd or even: With odd results go one step down, and even results go one step up.
After the zeros at a(0), a(2) and a(4) and -1 at a(1), the terms stay strictly positive for a long time, although from the terms of A257805 it can be seen that the sequence must again fall to the negative side somewhere between n = 541110611 and n = 1051158027 (i.e., A218600(33) .. A218600(34)). Indeed the fourth zero occurs at n = 671605896, and the second negative term right after that as a(671605897) = -1.
The maximum positive value reached prior to the slide into negative territory is 2614822 for a(278998626) and a(278998628). - Hans Havermann, May 23 2015

			We consider 0 to have no nonleading zeros, so first we get to 0 -> 0+1+0 = 1, and 1 is odd, so we go one step down from the starting value a(0)=0, and thus a(1) = -1.
1 has no nonleading zeros, so we get 1 -> 1+1+0 = 2, and 2 is even, so we go one step up, and thus a(2) = 0.
2 has one nonleading zero in binary "10", so we get 2 -> 2+1+1 = 4, and 4 is also even, so we go one step up, and thus a(3) = 1.
4 has two nonleading zeros in binary "100", so we get 4 -> 4+2+1 = 7, 7 is odd, so we go one step down, and thus a(4) = 0.

Antti Karttunen, Table of n, a(n) for n = 0..8727
Hans Havermann, Graph of 2*10^9 terms
Hans Havermann, Detail graph of the eventual crossover to negative terms and a listing of its associated zeros
Wikipedia, Blancmange curve

Cf. A218600, A233271, A233272, A257807, A257808, A257803, A257804, A257805.

Cf. also A218542, A218543, A218789 and A233270 (compare the scatter plots).

A070939 = n->#binary(n)+!n; \\ From M. F. Hasler
A080791 = n->if(n<1,0,(A070939(n)-hammingweight(n)));
A233272 = n->(n + A080791(n) + 1);
A257806_write_bfile(up_to_n) = { my(n,a_n=0,b_n=0); for(n=0, up_to_n, write("b257806.txt", n, " ", a_n); b_n = A233272(b_n); a_n += ((-1)^b_n)); };
A257806_write_bfile(8727);

def A257806_print_upto(n):
  a = 0
  b = 0
  for n in range(n):
     print(b, end=", ")
     ta = a
     c0 = 0
     while ta>0:
         c0 += 1-(ta&1)
         ta >>= 1
     a += 1 + c0
     b += ((2*(1-(a&1))) - 1)
# By Antti Karttunen after Alex Ratushnyak's Python-code for A216431.

(define (A257806 n) (- (A257808 n) (A257807 n)))

;; Using memoizing definec-macro.
(definec (A257806 n) (if (zero? n) n (+  (expt -1 (A233271 n)) (A257806 (- n 1)))))

a(n) = A257808(n) - A257807(n).
a(0) = 0; and for n >= 1, a(n) = a(n-1) + (-1)^A233271(n).
Other identities. For all n >= 0:
a(A218600(n+1)) = -A257805(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

A373123 Sum of all squarefree numbers from 2^(n-1) to 2^n - 1.

Original entry on oeis.org

1, 5, 18, 63, 218, 891, 3676, 15137, 60580, 238672, 953501, 3826167, 15308186, 61204878, 244709252, 979285522, 3917052950, 15664274802, 62663847447, 250662444349, 1002632090376, 4010544455838, 16042042419476, 64168305037147, 256675237863576

Offset: 1

Gus Wiseman, May 27 2024

nonn

			This is the sequence of row sums of A005117 treated as a triangle with row-lengths A077643:
   1
   2   3
   5   6   7
  10  11  13  14  15
  17  19  21  22  23  26  29  30  31
  33  34  35  37  38  39  41  42  43  46  47  51  53  55  57  58  59  61  62

Counting all numbers (not just squarefree) gives A010036.
For the sectioning of A005117:
Row-lengths are A077643, partial sums A143658.
First column is A372683, delta A373125, indices A372540, firsts of A372475.
Last column is A372889, delta A373126, indices A143658, diffs A077643.
For primes instead of powers of two:
- sum A373197
- length A373198 = A061398 - 1
- maxima A112925, opposite A112926
For prime instead of squarefree:
- sum A293697 (except initial terms)
- length A036378
- min A104080 or A014210, indices A372684 (firsts of A035100)
- max A014234, delta A013603
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers, first differences A076259.
A030190 gives binary expansion, reversed A030308.
A070939 or (preferably) A029837 gives length of binary expansion.
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
Cf. A029931, A048793, A049093, A049094, A059015, A069010, A077641.

Table[Total[Select[Range[2^(n-1),2^n-1],SquareFreeQ]],{n,10}]

a(n) = my(s=0); forsquarefree(i=2^(n-1), 2^n-1, s+=i[1]); s; \\ Michel Marcus, May 29 2024

A372889 Greatest squarefree number <= 2^n.

Original entry on oeis.org

1, 2, 3, 7, 15, 31, 62, 127, 255, 511, 1023, 2047, 4094, 8191, 16383, 32767, 65535, 131071, 262142, 524287, 1048574, 2097149, 4194303, 8388607, 16777214, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741822, 2147483647, 4294967295, 8589934591

Offset: 0

Gus Wiseman, May 27 2024

nonn

			The terms together with their binary expansions and binary indices begin:
      1:               1 ~ {1}
      2:              10 ~ {2}
      3:              11 ~ {1,2}
      7:             111 ~ {1,2,3}
     15:            1111 ~ {1,2,3,4}
     31:           11111 ~ {1,2,3,4,5}
     62:          111110 ~ {2,3,4,5,6}
    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}
   4094:    111111111110 ~ {2,3,4,5,6,7,8,9,10,11,12}
   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}

Positions of these terms in A005117 are A143658.
For prime instead of squarefree we have A014234, delta A013603.
For primes instead of powers of two we have A112925, opposite A112926.
Least squarefree number >= 2^n is A372683, delta A373125, indices A372540.
The opposite for prime instead of squarefree is A372684, firsts of A035100.
The delta (difference from 2^n) is A373126.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers, first differences A076259.
A030190 gives binary expansion, reversed A030308, length A070939 or A029837.
A061398 counts squarefree numbers between primes, exclusive.
A077643 counts squarefree terms between powers of 2, run-lengths of A372475.
Cf. A029931, A048793, A049093, A049094, A077641, A372433, A372472, A372473, A372541, A373197, A373198.

Table[NestWhile[#-1&,2^n,!SquareFreeQ[#]&],{n,0,15}]

a(n) = my(k=2^n); while (!issquarefree(k), k--); k; \\ Michel Marcus, May 29 2024

a(n) = A005117(A143658(n)).

a(n) = A070321(2^n). - R. J. Mathar, May 31 2024

cp's OEIS Frontend

A372540 Least k such that the k-th squarefree number has binary expansion of length n. Index of the smallest squarefree number >= 2^n.

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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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A372474 Least k such that the k-th prime number has exactly n zeros in its binary expansion.

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A059448 The parity of the number of zero digits when n is written in binary.

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A346633 Sum of even-indexed parts (even bisection) of the n-th composition in standard order.

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A372517 Least k such that the k-th prime number has exactly n ones in its binary expansion.

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A257806 a(n) = A257808(n) - A257807(n).

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