A214560 -id:A214560 - cp's OEIS frontend

A159918 Number of ones in binary representation of n^2.

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

Offset: 0

Reinhard Zumkeller, Apr 25 2009

The binary weight (A000120) of n^2.

a(n) = 0 iff n = 0. a(n) = 1 iff n = 2^k for some k >= 0. a(n) = 2 iff n = 3*2^k for some k >= 0. Szalay proves that a(n) = 3 iff n = 7*2^k, 23*2^k, or 2^a + 2^b for k >= 0 and a > b >= 0. It seems that a(n) = 4 iff n = 13*2^k, 15*2^k, 47*2^k, or 111*2^k but this has not been proven! Any other n with a(n) = 4 are greater than 10^50, and there are finitely many odd solutions. - Charles R Greathouse IV, Jan 20 2022

L. Szalay, The equations 2^n ± 2^m ± 2^l = z^2, Indagationes Mathematicae (N.S.) 13, no. 1 (2002), pp. 131-142.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Bernt Lindström, On the binary digits of a power, Journal of Number Theory, Volume 65, Issue 2, August 1997, Pages 321-324.
Nick MacKinnon, Problems and Solutions #12140, The American Mathematical Monthly, 126:9 (2019), 850.
K. B. Stolarsky, The binary digits of a power, Proc. Amer. Math. Soc. 71 (1978), 1-5.
Index entries for sequences related to binary expansion of n

Cf. A000120, A007088, A192085, A004159, A214560, A231897, A231898. For records see A230097.

a159918 = a000120 . a000290  -- Reinhard Zumkeller, Oct 12 2013

A159918 := proc(n) return add(b, b=convert(n^2, base, 2)): end: seq(A159918(n), n=0..100); # Nathaniel Johnston, Jun 23 2011

a[n_] := Total[IntegerDigits[n^2, 2]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 27 2021 *)

a(n)=hammingweight(n^2) \\ Charles R Greathouse IV, Aug 06 2015

def A159918(n):
    return bin(n*n).count('1') # Chai Wah Wu, Sep 03 2014

a(n) = A000120(A000290(n)); a(A077436(n)) = A000120(A077436(n)).
Lindström shows that lim sup wt(m^2)/log_2 m = 2. - N. J. A. Sloane, Oct 11 2013
a(n) = [x^(n^2)] (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018

A231897 a(n) = smallest m such that wt(m^2) = n (where wt(i) = A000120(i)), or -1 if no such m exists.

Original entry on oeis.org

0, 1, 3, 5, 13, 11, 21, 39, 45, 75, 155, 217, 331, 181, 627, 923, 1241, 2505, 3915, 5221, 6475, 11309, 15595, 19637, 31595, 44491, 69451, 113447, 185269, 244661, 357081, 453677, 1015143, 908091, 980853, 2960011, 4568757, 2965685, 5931189, 11862197, 20437147

Offset: 0

N. J. A. Sloane, Nov 19 2013

Conjecture: a(n) is never -1. (It seems likely that the arguments of Lindström (1997) could be modified to establish this conjecture.)

a(n) is the smallest m such that A159918(m) = n (or -1 if ...).

Hugo Pfoertner, Table of n, a(n) for n = 0..110 (terms 0..70 from Donovan Johnson, significant extension enabled by programs provided in Code Golf challenge).
Code Golf Stackexchange, Smallest and largest 100-bit square with maximum Hamming weight, fastest code challenge started Dec 15 2022.
Bernt Lindström, On the binary digits of a power, Journal of Number Theory, Volume 65, Issue 2, August 1997, Pages 321-324.

Cf. A000120, A159918, A230097, A211201, A231898, A214560.

A089998 are the corresponding squares.

a231897 n = head [x | x <- [1..], a159918 x == n]
-- Reinhard Zumkeller, Nov 20 2013

a(n)=if(n,my(k); while(hammingweight(k++^2)!=n,); k, 0) \\ Charles R Greathouse IV, Aug 06 2015

def wt(n): return bin(n).count('1')
def a(n):
    m = 2**(n//2) - 1
    while wt(m**2) != n: m += 1
    return m
print([a(n) for n in range(32)]) # Michael S. Branicky, Feb 06 2022

a(n) = 2*A211201(n-1) + 1 for n >= 1. - Hugo Pfoertner, Feb 06 2022

a(26)-a(40) from Reinhard Zumkeller, Nov 20 2013

A231898 a(n) = smallest k with property that for all m >= k, there is a square N^2 whose binary expansion contains exactly n 1's and m 0's; or -1 if no such k exists.

Original entry on oeis.org

-1, -1, 2, -1, 4, 3, 4, 3, 4, 5, 5, 5, 6, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

N. J. A. Sloane, Nov 19 2013

a(n) = -1 for n = 1, 2 and 4, because all squares with exactly 1, 2 or 4 1's in their binary expansion must contain an even number of 0's.
Conjecture: Apart from n=1, 2 and 4, no other a(n) is -1.
See A214560 for a related conjecture.

			Here is a table whose columns give:
N, N^2, number of bits in N^2, number of 1's in N^2, number of 0's in N^2:
0 0 1 0 1
1 1 1 1 0
2 4 3 1 2
3 9 4 2 2
4 16 5 1 4
5 25 5 3 2
6 36 6 2 4
7 49 6 3 3
8 64 7 1 6
9 81 7 3 4
10 100 7 3 4
11 121 7 5 2
12 144 8 2 6
13 169 8 4 4
14 196 8 3 5
15 225 8 4 4
16 256 9 1 8
17 289 9 3 6
18 324 9 3 6
19 361 9 5 4
...
a(n) is defined by the property that for all m >= a(n), the table contains a row ending n m. For example, there are rows ending 3 2, 3 3, 3 4, 3 5, ..., but not 3 1, so a(3) = 2.
a(5)=4: for t>=0, (11*2^t)^2 contains 5 1's and 2t+2 0's and (25*2^t)^2 contains 5 1's and 2t+5 0's, so for m >= 4 there is a number N such that N^2 contains 5 1's and m 0's. Also 4 is the smallest number with this property, so a(5) = 4.

Cf. A000120, A023416, A159918, A214560, A230097, A231897.

Missing word in definition supplied by Jon Perry, Nov 20 2013.

A214562 Number of 0's in binary expansion of n^n.

Original entry on oeis.org

0, 0, 2, 1, 8, 6, 10, 9, 24, 15, 23, 19, 34, 21, 31, 26, 64, 39, 49, 40, 61, 44, 63, 46, 95, 59, 82, 61, 98, 79, 97, 71, 160, 88, 112, 92, 129, 96, 115, 109, 160, 105, 131, 118, 178, 125, 159, 134, 228, 138, 178, 146, 207, 141, 183, 154, 245, 161, 192, 167, 231, 195

Offset: 0

Alex Ratushnyak, Jul 21 2012

Alois P. Heinz, Table of n, a(n) for n = 0..16383

Cf. A023416, A078565, A214560.

vector(66, n, b=binary((n-1)^(n-1)); sum(j=1, #b, 1-b[j])) /* Joerg Arndt, Jul 21 2012 */

for n in range(300):
   c = 0
   b = int(n**n)
   while b>0:
       c += 1-(b&1)
       b//=2
   print( c, end=", ")

a(n) = A023416(A000312(n)).

a(2^k) = k*2^k. - Chai Wah Wu, Dec 24 2024

cp's OEIS Frontend

A159918 Number of ones in binary representation of n^2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A231897 a(n) = smallest m such that wt(m^2) = n (where wt(i) = A000120(i)), or -1 if no such m exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

PARI

Python

Formula

Extensions

A231898 a(n) = smallest k with property that for all m >= k, there is a square N^2 whose binary expansion contains exactly n 1's and m 0's; or -1 if no such k exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A214562 Number of 0's in binary expansion of n^n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula