A003754 -id:A003754 - cp's OEIS frontend

A293430 Persistently squarefree numbers for base-2 shifting: Numbers n such that all terms in finite set [n, floor(n/2), floor(n/4), floor(n/8), ..., 1] are squarefree.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 23, 26, 29, 30, 31, 42, 43, 46, 47, 53, 58, 59, 61, 62, 85, 86, 87, 93, 94, 95, 106, 107, 118, 119, 122, 123, 170, 173, 174, 186, 187, 190, 191, 213, 214, 215, 237, 238, 239, 246, 247, 341, 346, 347, 349, 373, 374, 381, 382, 383, 426, 427, 429, 430, 431, 474, 478, 479

Offset: 1

Antti Karttunen and Michael De Vlieger, Oct 10 2017

nonn

Question: Is this sequence infinite? (My guess: yes). This is equivalent to questions asked in A293230. See also comments at A293441 and A293517.

For any odd n that is present, 2n is also present.

			For 479 we see that 479 is prime (thus squarefree, in A005117), [479/2] = 239 is also a prime, [239/2] = 119 = 7*17 (a squarefree composite), [119/2] = 59 (a prime), [59/2] = 29 (a prime), [29/2] = 14 = 2*7 (a squarefree composite), [14/2] = 7 (a prime), [7/2] = 3 (a prime), [3/2] = 1 (the end of halving process 1 is also squarefree), thus all the values obtained by repeated halving were squarefree and 479 is a member of this sequence. Here [ ] stands for taking floor.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Marked terms in the binary tree illustration of A293230.
Subsequence of A293427 (thus also of A003754 and of A005117).
Positions of nonzero terms in A293233.
Cf. A293441, A293517, A293523 (for floor(n/3^k) analog), A293437 (for a subsequence).

With[{s = Fold[Append[#1, MoebiusMu[#2] #1[[Floor[#2/2]]]] &, {1}, Range[2, 480]]}, Flatten@ Position[s, ?(# != 0 &)]] (* _Michael De Vlieger, Oct 10 2017 *)

is_persistently_squarefree(n,base) = { while(n>1, if(!issquarefree(n),return(0)); n \= base); (1); };
isA293430(n) = is_persistently_squarefree(n,2);
n=0; k=1; while(k <= 10000, n=n+1; if(isA293430(n),write("b293430.txt", k, " ", n);k=k+1)); \\ Antti Karttunen, Oct 11 2017

A280873 Numbers whose binary expansion does not begin 10 and does not contain 2 adjacent 0's; Ahnentafel numbers of X-chromosome inheritance of a male.

Original entry on oeis.org

0, 1, 3, 6, 7, 13, 14, 15, 26, 27, 29, 30, 31, 53, 54, 55, 58, 59, 61, 62, 63, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127, 213, 214, 215, 218, 219, 221, 222, 223, 234, 235, 237, 238, 239, 245, 246, 247, 250, 251, 253, 254, 255

Offset: 0

Floris Strijbos, Jan 09 2017

The number of ancestors at generation m from whom a living individual may have received an X chromosome allele is F_m, the m-th term of the Fibonacci Sequence.
From Antti Karttunen, Oct 11 2017: (Start)
The starting offset is zero (with a(0) = 0) for the same reason that we have A003714(0) = 0. Indeed, b(n) = A054429(A003714(n)) for n >= 0 yields the terms of this sequence, but in different order.
A163511(a(n)) for n >= 0 gives a permutation of squarefree numbers (A005117). See also A277006.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..10946
David Eppstein, Self-recursive generators (Python recipe)
L. A. D. Hutchison, N. M. Myres and S. R. Woodward, Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships, Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04, Colorado Springs), pp. 42-49, Sept. 2004.
Index entries for sequences related to binary expansion of n

Cf. A003714, A054429, A365538.
Intersection of A003754 and A004760.
Positions where A163511 obtains squarefree (A005117) values.
Cf. also A293437 (a subsequence).

gen[0]:= {0,1,3}:
gen[1]:= {6,7}:
for n from 2 to 10 do
  gen[n]:= map(t -> 2*t+1, gen[n-1]) union
      map(t -> 2*t, select(type, gen[n-1],odd))
od:
sort(convert(`union`(seq(gen[i],i=0..10)),list)); # Robert Israel, Oct 11 2017

male = {1, 3}; generations = 8;
Do[x = male[[i - 1]]; If[EvenQ[x],
                          male = Append[ male,   2*x + 1] ,
                          male = Flatten[Append[male, {2*x, 2*x + 1}]]]
       , {i, 3, Fibonacci[generations + 1]}]; male

isA003754(n) = { n=bitor(n, n>>1)+1; n>>=valuation(n, 2); (n==1); }; \\ After Charles R Greathouse IV's Feb 06 2017 code.
isA004760(n) = (n<2 || (binary(n)[2])); \\ This function also from Charles R Greathouse IV, Sep 23 2012
isA280873(n) = (isA003754(n) && isA004760(n));
n=0; k=0; while(k <= 10946, if(isA280873(n),write("b280873.txt", k, " ", n);k=k+1); n=n+1;); \\ Antti Karttunen, Oct 11 2017

def A280873():
    yield 1
    for x in A280873():
        if ((x & 1) and (x > 1)):
            yield 2*x
        yield 2*x+1
def take(n, g):
  '''Returns a list composed of the next n elements returned by generator g.'''
  z = []
  if 0 == n: return(z)
  for x in g:
    z.append(x)
    if n > 1: n = n-1
    else: return(z)
take(120, A280873())
# Antti Karttunen, Oct 11 2017, after the given Mathematica-code (by Floris Strijbos) and a similar generator-example for A003714 by David Eppstein (cf. "Self-recursive generators" link).

{a(n) : n >= 1} = {k >= 1 : A365538(A054429(k)) > 0}. - Peter Munn, Jan 22 2024

a(0) = 0 prepended and more descriptive alternative name added by Antti Karttunen, Oct 11 2017

A056973 Number of blocks of {0,0} in the binary expansion of n.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Eric Weisstein's World of Mathematics, Digit Block.
Index entries for sequences related to binary expansion of n

Cf. A003754, A014081, A023416, A033264, A037800.

Cf. A107782.

a056973 = f 0 where
   f y x = if x == 0 then y else f (y + 0 ^ (mod x 4)) $ div x 2
-- Reinhard Zumkeller, Mar 31 2015

f:= proc(n) option remember;
     if n mod 4 = 0 then 1 + procname(n/2)
     else procname(floor(n/2))
     fi
end proc:
f(1):= 0:
map(f, [$1..200]); # Robert Israel, Sep 02 2015

f[n_] := Count[Partition[IntegerDigits[n, 2], 2, 1], {0, 0}]; Table[f@ n, {n, 0, 102}] (* Michael De Vlieger, Sep 01 2015, after Robert G. Wilson v at A014081 *)
SequenceCount[#,{0,0},Overlaps->True]&/@(IntegerDigits[#,2]&/@Range[0,120]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 24 2018 *)

a(n) = { my(x = bitor(n, n>>1));
         if (x == 0, 0, 1 + logint(x, 2) - hammingweight(x)) }
vector(102, i, a(i))  \\ Gheorghe Coserea, Sep 01 2015

a(2n) = a(n) + [n is even], a(2n+1) = a(n).
G.f.: 1/(1-x) * Sum_{k>=0} t^4/((1+t)*(1+t^2)) where t=x^(2^k). - Ralf Stephan, Sep 10 2003
a(n) = A023416(n) - A033264(n). - Ralf Stephan, Sep 10 2003
Sum_{n>=1} a(n)/(n*(n+1)) = 2 - 3*log(2)/2 - Pi/4 (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

A333630 Least STC-number of a composition whose sequence of run-lengths has STC-number n.

Original entry on oeis.org

0, 1, 3, 5, 7, 14, 11, 13, 15, 30, 43, 29, 23, 46, 27, 45, 31, 62, 122, 61, 87, 117, 59, 118, 47, 94, 107, 93, 55, 110, 91, 109, 63, 126, 250, 125, 343, 245, 123, 246, 175, 350, 235, 349, 119, 238, 347, 237, 95, 190, 378, 189, 215, 373, 187, 374, 111, 222, 363

Offset: 0

Gus Wiseman, Mar 31 2020

nonn

All terms belong to A003754.

A composition of n is a finite sequence of positive integers summing to n. The composition with STC-number k (row k of 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.

			The sequence together with the corresponding compositions begins:
   0: ()
   1: (1)
   3: (1,1)
   5: (2,1)
   7: (1,1,1)
  14: (1,1,2)
  11: (2,1,1)
  13: (1,2,1)
  15: (1,1,1,1)
  30: (1,1,1,2)
  43: (2,2,1,1)
  29: (1,1,2,1)
  23: (2,1,1,1)
  46: (2,1,1,2)
  27: (1,2,1,1)
  45: (2,1,2,1)
  31: (1,1,1,1,1)
  62: (1,1,1,1,2)

Position of first appearance of n in A333627.
All of the following pertain to compositions in standard order (A066099):
- The length is A000120.
- Compositions without terms > 2 are A003754.
- Compositions without ones are ranked by A022340.
- The partial sums from the right are A048793.
- The sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Equal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- The partial sums from the left are A272020.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- Runs-resistance is A333628.
- First appearances of run-resistances are A333629.
Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
seq=Table[Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2,{n,0,1000}];
Table[Position[seq,i][[1,1]],{i,First[Split[Union[seq],#1+1==#2&]]}]-1

A107345 From the binary representation of n: binomial(number of zeros, number of blocks of contiguous zeros).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 3, 3, 2, 3, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 4, 6, 3, 6, 3, 3, 2, 6, 3, 1, 1, 3, 1, 1, 1, 4, 3, 3, 2, 3, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 6, 5, 10, 4, 10, 6, 6, 3, 10, 6, 4, 3, 6, 3, 3, 2, 10, 6, 4, 3, 4, 1, 1, 1, 6, 3, 1, 1, 3, 1, 1, 1, 5, 4, 6, 3, 6, 3, 3

Offset: 0

Reinhard Zumkeller, May 23 2005

nonn

a(n) = binomial(A023416(n), A087116(n)); a(A003754(n)) = 1.
First occurrence of k: 1, 4, 8, 16, 32, 34, 128, 256, 512, 66, 2048, 4096, 8192, 16384, 130, 65536, 131072, 262144, 524288, 266, 258, ..., . k must occur by 2^k. - Robert G. Wilson v
Record values: 1, 4, 8, 16, 32, 34, 66, 130, 258, 514, 522, 1026, 1034, 2058, 4106, 4138, 8202, 8234, 16394, 16426, 32810, 65578, 65706, 131114, 131242, 262186, 262314, 524458, 1048746, 1049258, 2097322, 2097834, 4194474, 4194986, 8389290, 8391338, ..., . - Robert G. Wilson v

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A107782, A136277.

Cf. A007318, A023416, A087116.

a107345 n = a007318' (a023416 n) (a087116 n)
-- Reinhard Zumkeller, Mar 31 2015

f[n_] := Block[{id = IntegerDigits[n, 2]}, Binomial[ Count[id, 0], Floor[(Length@ Split@ id + 1)/2]]]; Table[f@n, {n, 0, 102}] (* Robert G. Wilson v, Apr 01 2008 *)

A107782 In binary representation of n: (number of zeros) minus (number of blocks of contiguous zeros).

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, May 25 2005

nonn

a(n) = A023416(n) - A087116(n); a(A003754(n)) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A107345.
Cf. A056973. - R. J. Mathar, Aug 24 2008
Cf. A023416, A087116, A003754.

a107782 n = a023416 n - a087116 n  -- Reinhard Zumkeller, Mar 31 2015

A181631 Triangle by rows, number of leading 1's in the maximal Fibonacci representation (A104326).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 1, 2, 2, 3, 4, 5, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7, 8

Offset: 1

Gary W. Adamson, Nov 02 2010

Row sums = A001911: (1, 3, 6, 11, 19, 32, ...).
A112310 = number of 1's in the maximal Fibonacci representation, which has headings of (..., 8, 5, 3, 2, 1) filling entries from the right to left; as opposed to the minimal Fibonacci representation (A014417) which starts from the left. For example, 8 in maximal = 1011 = (5 + 2 + 1) whereas in minimal = (10000) = (8).
Rows have (1, 2, 3, 5, 8, ...) terms.

			First few rows of the triangle:
  1;
  1, 2;
  1, 2, 3;
  1, 1, 2, 3, 4;
  1, 1, 1, 2, 2, 3, 4, 5;
  1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6;
  ...
Example: a(14) = 1 since 14 in the maximal Fibonacci representation is 10111.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045 (row lengths), A003754, A001911 (row sums), A014417, A090996, A104326, A112310.

f[s_] := Module[{i = FirstPosition[s, 0]}, If[MissingQ[i], Length[s], i[[1]] - 1]]; f /@ Select[IntegerDigits[#, 2] & /@ Range[300], SequencePosition[#, {0, 0}] == {} &] (* Amiram Eldar, May 31 2025 *)

a(n) = A090996(A003754(n+1)). - Amiram Eldar, May 31 2025

More terms from Amiram Eldar, May 31 2025

A196168 In binary representation of n: replace each 0 with 1, and each 1 with 10.

Original entry on oeis.org

1, 2, 5, 10, 11, 22, 21, 42, 23, 46, 45, 90, 43, 86, 85, 170, 47, 94, 93, 186, 91, 182, 181, 362, 87, 174, 173, 346, 171, 342, 341, 682, 95, 190, 189, 378, 187, 374, 373, 746, 183, 366, 365, 730, 363, 726, 725, 1450, 175, 350, 349, 698, 347, 694, 693, 1386

Offset: 0

Reinhard Zumkeller, Oct 28 2011

nonn

All terms are numbers with no two adjacent zeros in binary representation, cf. A003754;
a(odd) = even and a(even) = odd;
A023416(a(n)) <= A000120(a(n)), equality iff n = 2^k - 1 for k > 0;
A055010(n+1) = A196168(A000079(n));
A000120(a(n)) = A070939(n);
A023416(a(n)) = A000120(n);
A070939(a(n)) = A070939(n) + A000120(n).

			n =  7 ->  111 ->  101010 ->  a(7) = 42;
n =  8 -> 1000 ->   10111 ->  a(8) = 23;
n =  9 -> 1001 ->  101110 ->  a(9) = 46;
n = 10 -> 1010 ->  101101 -> a(10) = 45;
n = 11 -> 1011 -> 1011010 -> a(11) = 90;
n = 12 -> 1100 ->  101011 -> a(12) = 43.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A179888, A005614.

import Data.List (unfoldr)
a196168 0 = 1
a196168 n = foldl (\v b -> (2 * v + 1)*(b + 1)) 0 $ reverse $ unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2) n
   where r v b = (2 * v + 1)*(b+1)

Table[FromDigits[Flatten[IntegerDigits[n,2]/.{{0->1,1->{1,0}}}],2],{n,0,120}] (* Harvey P. Dale, Dec 12 2017 *)

def a(n):
    b = bin(n)[2:]
    return int(b.replace('1', 't').replace('0', '1').replace('t', '10'), 2)
print([a(n) for n in range(56)]) # Michael S. Branicky, Oct 28 2021

n = Sum_{i=0..1} b(i)*2^i with 0 <= b(i) <= 1, L >= 0, then a(n) = h(0,L) with h(v,i) = if i > L then v, otherwise h((2*v+1)*(b(i)+1),i-1).
From Jeffrey Shallit, Oct 28 2021: (Start)
a(n) satisfies the recurrences:
a(2n+1) = 2*a(2n)
a(4n) = -2*a(n) + 3*a(2n)
a(8n+2) = -8*a(n) + 8*a(2n) + a(4n+2)
a(8n+6) = -4*a(2n) + 5*a(4n+2)
which shows that a(n) is a 2-regular sequence. (End)

A293437 Numbers n that are persistently squarefree for base-2 shifting (in A293430), and for which A163511(n) is also in A293430.

Original entry on oeis.org

1, 3, 6, 7, 13, 14, 15, 26, 29, 30, 31, 58, 62, 239, 247, 478, 479, 494, 958, 245757, 491514

Offset: 1

Antti Karttunen, Oct 11 2017

The motivation for this sequence is the observation that one of the necessary conditions for inclusion in A293430 is almost the same as for what is required from k that A163511(k) were squarefree. Namely, all terms of A293430 can be found in A003754 (the former is a subsequence of the latter), while A163511(k) yields a squarefree number if and only if k is in A280873, which is a subsequence of A003754 (actually its intersection with A004760). Thus this sequence must be in the intersection of A293430 and A004760, which implies that the binary expansion of all terms is free of adjacent 0's and furthermore, none begins with bits "10". Indeed, in base-2 the terms look as: 1, 11, 110, 111, 1101, 1110, 1111, 11010, 11101, 11110, 11111, 111010, 111110, 11101111, 11110111, 111011110, 111011111, 111101110, 1110111110, 111011111111111101, 1110111111111111010.
A163511 applied to the first 21 terms yields 2, 3, 6, 5, 15, 10, 7, 30, 21, 14, 11, 42, 22, 187, 119, 374, 247, 238, 494, 6837, 13674, that in binary look like: 10, 11, 110, 101, 1111, 1010, 111, 11110, 10101, 1110, 1011, 101010, 10110, 10111011, 1110111, 101110110, 11110111, 11101110, 111101110, 1101010110101, 11010101101010. These are of course numbers such that both n and A243071(n) are in A293430, but not listed in ascending order.
Question: Is this sequence finite?
See also the binary tree illustration in A293230.

			For n = 245757 which itself is squarefree (as 245757 = 3*81919) applying the map x -> floor(x/2) iteratively down to 1 yields a finite sequence 122878, 61439, 30719, 15359, 7679, 3839, 1919, 959, 479, 239, 119, 59, 29, 14, 7, 3, 1, whose terms are all squarefree also. Moreover, A163511(245757) = 6837 = 3*43*53, a squarefree number too (this is already guaranteed by the fact that the two most significant bits in base-2 expansion of 245757 are both 1's). Applying the same approximate halving map iteratively yields now the sequence: 3418, 1709, 854, 427, 213, 106, 53, 26, 13, 6, 3, 1, and also here every term is squarefree. Thus 245757 is included in this sequence.

Cf. A003754, A004760, A005117, A163511, A243071, A280873, A293230, A293233, A293430.

default(primelimit,(2^31)+(2^30));
is_persistently_squarefree(n,base) = { while(n>1, if(!issquarefree(n),return(0)); n \= base); (1); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A054429(n) = ((3<<#binary(n\2))-n-1); \\ After M. F. Hasler, Aug 18 2014
A163511(n) = if(!n,1,A005940(1+A054429(n)));
isA293430(n) = is_persistently_squarefree(n,2);
n=0; k=1; while(n <= 2^26, n=n+1; if(isA293430(n)&&isA293430(A163511(n)),write("b293437.txt", k, " ", n);k=k+1));

;; With Antti Karttunen's IntSeq-library.
(define A293437 (MATCHING-POS 1 1 (lambda (n) (and (not (zero? (A293233 n))) (not (zero? (A293233 (A163511 n))))))))

n is present if and only if A293233(n)*A293233(A163511(n)) <> 0.

A331362 a(n) is the greatest value of the form s_1 + ... + s_k such that the concatenation of the binary representations of s_1^2, ..., s_k^2 equals the binary representation of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 3, 3, 4, 3, 5, 3, 4, 4, 4, 4, 5, 4, 5, 3, 4, 6, 4, 4, 5, 3, 4, 3, 4, 4, 4, 4, 5, 5, 7, 5, 6, 4, 4, 4, 5, 4, 6, 4, 5, 5, 5, 5, 6, 8, 5, 5, 6, 4, 4, 4, 5, 6, 7, 4, 5, 5, 5, 5, 6, 5, 9, 4, 5, 4, 4, 4

Offset: 0

Rémy Sigrist, Jan 14 2020

As 0 and 1 are squares, we can always split the binary representation of a number into squares, and the sequence is well defined.

			For n = 8:
- the binary representation of 8 is "1000",
- we can split it into "100" and "0" (2^2 and 0^2),
- or into "1" and "0" and "0" and "0" (1^2 and 0^2 and 0^2 and 0^2),
- so a(8) = max(2+0, 1+0+0+0) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Rémy Sigrist, Density plot of the first 2^20 terms
Rémy Sigrist, PARI program for A331362
Index entries for sequences related to binary expansion of n

Cf. A000120, A003754.

PARI
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See Links section.
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a(n) >= A000120(n) with equality iff n belongs to A003754.

a(n^2) = n.

cp's OEIS Frontend

A293430 Persistently squarefree numbers for base-2 shifting: Numbers n such that all terms in finite set [n, floor(n/2), floor(n/4), floor(n/8), ..., 1] are squarefree.

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A280873 Numbers whose binary expansion does not begin 10 and does not contain 2 adjacent 0's; Ahnentafel numbers of X-chromosome inheritance of a male.

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A056973 Number of blocks of {0,0} in the binary expansion of n.

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A333630 Least STC-number of a composition whose sequence of run-lengths has STC-number n.

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A107345 From the binary representation of n: binomial(number of zeros, number of blocks of contiguous zeros).

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A107782 In binary representation of n: (number of zeros) minus (number of blocks of contiguous zeros).

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A181631 Triangle by rows, number of leading 1's in the maximal Fibonacci representation (A104326).

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A196168 In binary representation of n: replace each 0 with 1, and each 1 with 10.

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