A094694 -id:A094694 - cp's OEIS frontend

A094695 Numbers having in binary representation fewer ones than in their squares.

5, 9, 10, 11, 13, 17, 18, 19, 20, 21, 22, 25, 26, 27, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 93, 97, 98, 99

Offset: 1

Reinhard Zumkeller, May 20 2004

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

Cf. A000120, A001737, A007088, A077436, A094694, A159918.

Select[Range[100],DigitCount[#,2,1]Harvey P. Dale, May 29 2017 *)

A000120(a(n)) < A000120(A000290(a(n))).

A232243 a(n) = wt(n^2) - wt(n), where wt(n) = A000120(n) is the binary weight function.

Original entry on oeis.org

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

Offset: 0

Jon Perry, Nov 20 2013

A077436 lists n for which a(n) = 0.

A094694 lists n for which a(n) < 0.

			a(5): 5 = 101_2, 25 = 11001_2, so a(5) = 3 - 2 = 1.
a(23): 23 = 10111_2, 529 = 10001001_2, so a(23) = 3 - 4 = -1.

Dumitru Damian, Table of n, a(n) for n = 0..10000

Cf. A000120, A159918, A077436, A094694, A231898.

function bitCount(n) {
   var i,c,s;
   c=0;
   s=n.toString(2);
   for (i=0;i

a(n) = hammingweight(n^2) - hammingweight(n); \\ Michel Marcus, Mar 05 2023

def A232243(n): return (n**2).bit_count()-n.bit_count()
print(list(A232243(n) for n in range(10**2)))  # Dumitru Damian, Mar 04 2023

a(n) = A159918(n) - A000120(n).

A236514 Primes with a binary weight greater than or equal to the binary weight of their squares.

Original entry on oeis.org

2, 3, 7, 23, 31, 47, 79, 127, 157, 191, 223, 317, 367, 379, 383, 479, 727, 751, 887, 1087, 1151, 1277, 1279, 1451, 1471, 1531, 1663, 1783, 1789, 1951, 2297, 2557, 2927, 3067, 3259, 3319, 3581, 3583, 3967, 4253, 4349, 5119, 5231, 5503, 5807, 5821, 6079, 6143, 6271, 6653, 6871, 6911, 7039, 7103, 7151

Offset: 1

Irina Gerasimova, Jan 27 2014

Primes p such that A000120(p) = A000120(p^2): 2, 3, 7, 31, 79, 127, 157, 317, 379, 751, 1087, 1151, 1277, 1279,...

			2 is in this sequence because 2 is 10 in binary representation, and it has as many 1s as its square 4, which is 100 in binary.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A077436, A094694.

bc[n_] := DigitCount[n, 2][[1]]; Select[Range[7151], PrimeQ[#] && bc[#] >= bc[#^2] &] (* Giovanni Resta, Jan 28 2014 *)
Select[Prime[Range[1000]], DigitCount[#, 2, 1] >= DigitCount[#^2, 2, 1] &] (* Alonso del Arte, Jan 28 2014 *)

is(n)=hammingweight(n^2)<=hammingweight(n) && isprime(n) \\ Charles R Greathouse IV, Mar 18 2014

Primes p such that A000120(p) >= A000120(p^2).

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

A363799 Numbers whose binary representation has more 1-bits than its cube.

Original entry on oeis.org

407182835067, 445317119867, 478351981947, 814365670134, 873268508637, 890634239734, 956703963894, 956703964539, 1628731340268, 1746537017274, 1781268479468, 1913407927788, 1913407929078, 2774213097787, 3257462680536, 3493074034548, 3562536958936, 3573277243773

Offset: 1

Zhao Hui Du, Jun 23 2023

a(n) must have more 1-bits than a(n)^3 when they are written in binary.

			407182835067 is a term because A000120(407182835067) = 29, while A192085(407182835067) = A000120(407182835067^3) = 28.

Chris K. Caldwell and G. L. Honaker, Jr., 445317119867, Prime Curios!
K. G. Hare, S. Laishram, and T. Stoll, Stolarsky's conjecture and the sum of digits of polynomial values, arXiv:1001.4169 [math.NT], 2010. See p. 3.
Thomas Stoll, On Stolarsky's conjecture: The sum of digits of n and n^h, Slides, 2010.

Cf. A000120, A192085, A138597 (equality).

Cf. A094694 (for squares).

isok(k) = hammingweight(k) > hammingweight(k^3); \\ Michel Marcus, Aug 07 2023

a(9)-a(18) from Martin Ehrenstein, Jul 31 2023

A293655 Numbers having in binary representation more zeros than their squares.

Original entry on oeis.org

181, 5221, 11309, 19637, 21577, 22805, 43151, 69451, 74969, 76845, 82709, 83539, 85029, 86283, 86581, 91205, 148245, 165013, 165061, 165418, 166027, 170021, 172213, 172615, 173095, 173101, 173162, 173331, 180405, 182433, 184587, 184885, 185363, 201829, 282713

Offset: 1

Alex Ratushnyak, Feb 06 2018

			181 in base 2 is 10110101, with 3 zeros, and 181^2 is 111111111111001, with 2 zeros.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A023416, A094694.

Select[Range[3*10^5], DigitCount[#, 2, 0] > DigitCount[#^2, 2, 0] &] (* Michael De Vlieger, Feb 21 2018 *)

nbz(n) = my(b=binary(n)); #b - hammingweight(n);
isok(n) = nbz(n) > nbz(n^2); \\ Michel Marcus, Feb 12 2018

def count0(n):
    return bin(n)[2:].count('0')
for n in range(1000000):
    if count0(n*n) < count0(n):
        print(str(n), end=',')

A352291 Odd numbers k such that hammingweight(k^2) < hammingweight(k).

Original entry on oeis.org

23, 47, 95, 111, 191, 223, 367, 383, 415, 447, 479, 727, 767, 831, 887, 895, 959, 1451, 1471, 1503, 1535, 1663, 1727, 1775, 1783, 1791, 1855, 1917, 1919, 1983, 2527, 2911, 2943, 2991, 3071, 3327, 3455, 3549, 3551, 3567, 3575, 3583, 3695, 3711, 3837, 3839, 3967, 3999, 5793, 5823, 5855, 5883, 5885, 5887, 5949, 5951, 5983, 5993, 5999

Offset: 1

Joerg Arndt, Mar 11 2022

Odd terms in A094694.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000120, A094694.

select(t -> convert(convert(t^2,base,2),`+`) < convert(convert(t,base,2),`+`), [seq(i,i=1..10^4,2)]); # Robert Israel, Mar 13 2022

Select[Range[1, 6000, 2], Greater @@ DigitCount[{#, #^2}, 2, 1] &] (* Amiram Eldar, Mar 11 2022 *)

forstep(n=1,10^4,2,if(hammingweight(n^2)

def ok(n): return n%2 == 1 and bin(n).count('1') > bin(n**2).count('1')
print([k for k in range(6000) if ok(k)]) # Michael S. Branicky, Mar 11 2022

A232245 Sum of the number of ones in binary representation of n and n^2.

Original entry on oeis.org

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

Offset: 0

Jon Perry, Nov 20 2013

The sequence is never 1 or 3, but seems to take on all other values. The fact it is never 3 can be used to prove if n^2 has exactly 4 1's then it must have an even number of 0's (A231898).

			5 is 101 and 25 is 11001, so a(5) = 2 + 3 = 5.

Cf. A000120, A159918, A077436, A094694, A231898, A232243.

function bitCount(n) {
var i,c,s;
c=0;
s=n.toString(2);
for (i=0;i

a(n) = A159918(n) + A000120(n).

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A094695 Numbers having in binary representation fewer ones than in their squares.

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A232243 a(n) = wt(n^2) - wt(n), where wt(n) = A000120(n) is the binary weight function.

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A236514 Primes with a binary weight greater than or equal to the binary weight of their squares.

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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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A363799 Numbers whose binary representation has more 1-bits than its cube.

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A293655 Numbers having in binary representation more zeros than their squares.

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A352291 Odd numbers k such that hammingweight(k^2) < hammingweight(k).

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A232245 Sum of the number of ones in binary representation of n and n^2.