A231897 -id:A231897 - 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

A089998 Smallest square with Hamming weight n (i.e., with exactly n 1's when written in binary).

Original entry on oeis.org

0, 1, 9, 25, 169, 121, 441, 1521, 2025, 5625, 24025, 47089, 109561, 32761, 393129, 851929, 1540081, 6275025, 15327225, 27258841, 41925625, 127893481, 243204025, 385611769, 998244025, 1979449081, 4823441401, 12870221809, 34324602361

Offset: 0

Reinhard Zumkeller, Nov 20 2003

A000120(a(n)) = n.

Donovan Johnson, Table of n, a(n) for n = 0..60

Cf. A000290, A089999, A061712, A090001, A231897.

a = Table[0, {30}]; Do[c = Count[IntegerDigits[n^2, 2], 1]; If[ a[[c + 1]] == 0, a[[c + 1]] = n^2; Print[c, " = ", n^2]], {n, 1, 360000}] (* Robert G. Wilson v, Dec 03 2003 *)
Join[{0},With[{s=DigitCount[Range[400000]^2,2,1]},Flatten[Table[ Position[ s,?(#==n&),1,1],{n,30}]]]^2] (* _Harvey P. Dale, Mar 03 2013 *)

a(n) = A231897(n)^2. - Hugo Pfoertner, Dec 27 2022

More terms from Robert G. Wilson v, Dec 03 2003

Offset corrected by Donovan Johnson, May 01 2012

A230097 Indices of records in A159918.

Original entry on oeis.org

0, 1, 3, 5, 11, 21, 39, 45, 75, 155, 181, 627, 923, 1241, 2505, 3915, 5221, 6475, 11309, 15595, 19637, 31595, 44491, 69451, 113447, 185269, 244661, 357081, 453677, 908091, 980853, 2960011, 2965685, 5931189, 11862197, 20437147, 22193965, 43586515, 57804981, 157355851

Offset: 1

N. J. A. Sloane, Oct 11 2013

The records themselves are not so interesting: 0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, ... (A357304).

Lindström mentions that the record value 34 in A159918 is first reached at n = 980853.

Bert Dobbelaere, Table of n, a(n) for n = 1..80, (terms 41..64 from Donovan Johnson, 65..70 from Hugo Pfoertner, missing 68 and 72..80 from Bert Dobbelaere).
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, A231897, A357304, A357658.

a230097 n = a230097_list !! (n-1)
a230097_list = 0 : f 0 0 where
   f i m = if v > m then i : f (i + 1) v else f (i + 1) m
           where v = a159918 i
-- Reinhard Zumkeller, Oct 12 2013
(Python 3.10+)
from itertools import count, islice
def A230097_gen(): # generator of terms
    c = -1
    for n in count(0):
        if (m := (n**2).bit_count())>c:
            yield n
            c = m
A230097_list = list(islice(A230097_gen(),20)) # Chai Wah Wu, Oct 01 2022

Lindström shows that lim sup wt(m^2)/log_2 m = 2.

a(19)-a(40) from Reinhard Zumkeller, Oct 12 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.

A351149 a(n) is the least exponent k such that the Hamming weight of n^(k+1) is not greater than the Hamming weight of n^k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 3, 3, 5, 1, 7, 1, 1, 1, 3, 3, 4, 3, 2, 5, 1, 1, 4, 7, 5, 1, 5, 1, 1, 1, 3, 3, 7, 3, 4, 4, 3, 3, 5, 2, 5, 5, 3, 1, 1, 1, 5, 4, 7, 7, 2, 5, 3, 1, 3, 5, 2, 1, 3, 1, 1, 1, 3, 3, 4, 3, 5, 7, 3, 3, 3, 4, 3, 4, 3, 3, 1, 3, 5, 5, 3, 2, 3, 5, 11

Offset: 1

Hugo Pfoertner, Feb 07 2022

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000120, A094694, A211201, A231897.

a[n_] := Module[{k = 1}, While[DigitCount[n^k, 2, 1] < DigitCount[n^(k + 1), 2, 1], k++]; k]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

for(n=1,87, for(k=1,oo, my(hw1=hammingweight(n^k), hw2=hammingweight(n^(k+1))); if(hw2<=hw1, print1(k,", "); break)))

def A351149(n):
    k = 1
    while bin(n**k)[2:].count("1") < bin(n**(k+1))[2:].count("1"): k += 1
    return(k)
print([A351149(n) for n in range(1, 88)]) # Karl-Heinz Hofmann, Feb 07 2022

A358702 a(n) is the least k > 0 such that the sum of the decimal digits of k^2 is n, or 0 if no such k exists.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 4, 0, 3, 8, 0, 0, 7, 0, 0, 13, 0, 24, 17, 0, 0, 43, 0, 0, 67, 0, 63, 134, 0, 0, 83, 0, 0, 167, 0, 264, 314, 0, 0, 313, 0, 0, 707, 0, 1374, 836, 0, 0, 1667, 0, 0, 2236, 0, 3114, 4472, 0, 0, 6833, 0, 0, 8167, 0, 8937, 16667, 0, 0, 21886, 0, 0, 29614

Offset: 1

Hugo Pfoertner, Dec 04 2022

The nonzero terms are A067179.

Cf. A056991, A231897 (similar for binary weight).

A359091 a(n) is the index of the smallest n-gonal number with binary weight n.

Original entry on oeis.org

6, 13, 9, 10, 24, 58, 34, 55, 67, 151, 134, 187, 201, 691, 350, 623, 1082, 1870, 2302, 3171, 5017, 13863, 13230, 6663, 24357, 50397, 35604, 60347, 63810, 107019, 181517, 365595, 624858, 1345485, 1002585, 1969415, 1191179, 7651731, 4592173, 7279863, 7403686, 17923182

Offset: 3

Ilya Gutkovskiy, Dec 16 2022

Eric Weisstein's World of Mathematics, Polygonal Number
Index entries for sequences related to binary expansion of n

Cf. A000120, A211201, A231897, A359092.

p[n_, k_] := (n - 2)*k*(k - 1)/2 + k; a[n_] := Module[{k = 1}, While[DigitCount[p[n, k], 2, 1] != n, k++]; k]; Array[a, 30, 3] (* Amiram Eldar, Dec 17 2022 *)

A386242 a(n) is the least perfect power A001597 with binary weight n.

Original entry on oeis.org

1, 9, 25, 27, 121, 125, 1521, 2025, 5625, 24025, 42875, 59319, 32761, 393129, 851929, 1540081, 6275025, 15327225, 27258841, 41925625, 127893481, 243204025, 385611769, 268336125, 1979449081, 4823441401, 12870221809, 25698491351, 51354402813, 127506840561, 205822820329

Offset: 1

Hugo Pfoertner, Jul 23 2025

Cf. A000120, A001597, A231897.

upto = 10^11; L = Table[2 upto, {2 + Log2@ upto}]; Do[n = 1; While[(v = n^k) <= upto, nb = Plus @@ IntegerDigits[v, 2]; If[L[[nb]] > v, L[[nb]] = v]; n++], {k, 2, Log2[upto]}]; Take[L, Position[L, 2 upto][[1, 1]] - 1] (* Giovanni Resta, Jul 23 2025 *)

a(27)-a(31) from Giovanni Resta, Jul 23 2025

cp's OEIS Frontend

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

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A089998 Smallest square with Hamming weight n (i.e., with exactly n 1's when written in binary).

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A230097 Indices of records in A159918.

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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.

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A351149 a(n) is the least exponent k such that the Hamming weight of n^(k+1) is not greater than the Hamming weight of n^k.

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A358702 a(n) is the least k > 0 such that the sum of the decimal digits of k^2 is n, or 0 if no such k exists.

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A359091 a(n) is the index of the smallest n-gonal number with binary weight n.

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Mathematica

A386242 a(n) is the least perfect power A001597 with binary weight n.

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