A192085 -id:A192085 - 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

A286375 a(n) = A278222(n^3).

Original entry on oeis.org

1, 2, 2, 36, 2, 96, 36, 840, 2, 1260, 96, 1260, 36, 2310, 840, 5040, 2, 1260, 1260, 69300, 96, 4620, 1260, 4320, 36, 1680, 2310, 12600, 840, 36960, 5040, 30240, 2, 1260, 1260, 831600, 1260, 415800, 69300, 2646000, 96, 27720, 4620, 900900, 1260, 221760, 4320, 277200, 36, 83160, 1680, 13860, 2310, 180180, 12600, 166320, 840, 15315300, 36960, 13860, 5040

Offset: 0

Antti Karttunen, May 09 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A000578, A278222, A286243, A286375, A286376.

Cf. A192085 (one of the matched sequences).

(define (A286375 n) (A278222 (* n n n)))

a(n) = A278222(A000578(n)) = A278222(n^3).

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

A192086 Numbers n such that the binary weight (A000120) of n^3 is less than the binary weight of n^2.

Original entry on oeis.org

219, 309, 349, 438, 467, 565, 618, 665, 698, 701, 817, 876, 885, 934, 1130, 1141, 1143, 1236, 1291, 1297, 1299, 1330, 1396, 1397, 1402, 1435, 1629, 1634, 1659, 1752, 1755, 1770, 1771, 1773, 1868, 1905, 2155, 2229, 2260, 2282, 2286, 2353, 2401, 2472, 2582

Offset: 1

Carl R. White, Jun 23 2011

Numbers n such that A192085(n) < A159918(n).

This sequence is infinite, since if k is in the sequence then so is 2k. - Charles R Greathouse IV, Sep 27 2016

			219^2 decimal = 1011101101011001 binary, and 219^3 decimal = 101000000100010100100011; Since the cube has fewer ones in its binary expansion than the square (eight versus ten), 219 is in the list.

Nathaniel Johnston, Table of n, a(n) for n = 1..2500

Cf. A000120, A159918, A192085.

[ n: n in [0..2600] | &+Intseq(n^3,2) lt &+Intseq(n^2,2) ];  // Bruno Berselli, Jun 24 2011

A192086 := proc(n) return `if`(add(b,b=convert(n^3,base,2))A192086(n),n=0..3000); # Nathaniel Johnston, Jun 23 2011

Select[Range[2800],DigitCount[#^3,2,1]Harvey P. Dale, Jun 30 2011 *)
Flatten@Position[Flatten@(Differences@DigitSum[#^2*{1, #}, 2] & /@Range@15000), ?(# < 0 &)] (* _Hans Rudolf Widmer, Aug 05 2024 *)

is(n)=hammingweight(n^3)Charles R Greathouse IV, Sep 27 2016

A234218 Primes whose cubes are odious.

Original entry on oeis.org

2, 13, 23, 29, 43, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 131, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 229, 241, 251, 293, 311, 313, 337, 353, 367, 383, 389, 397, 409, 419, 431, 439, 443, 461, 467, 479, 499, 509, 521, 541, 563, 577, 601

Offset: 1

Irina Gerasimova, Dec 21 2013

Primes p with odious p^3.
Note: "odious" means having an odd number of 1-bits in number's binary representation. So, put in another way, primes p such that A010060(A000578(p)) = 1. - Antti Karttunen, Dec 22 2013
Subsequence of the numbers 1, 2, 4, 8, 13, 16, 23, 25, 26, 29, 32, 35, 43, 45, 46, ... which have odious cubes.

			Prime 2 is in this sequence because 2^3 = 8 and 8 is odious number. Prime 13 is in this sequence because 13*3 = 2197 and 2197 is odious number.

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A000069, A000120, A000578, A010060, A125498, A192085, A194991.

Select[Prime[Range[1000]], OddQ[DigitCount[#^3, 2, 1]] &] (* Indranil Ghosh, Apr 02 2017 *)

is(n)=isprime(n) && hammingweight(n^3)%2 \\ Charles R Greathouse IV, Mar 17 2014

Missing terms added by Antti Karttunen, Dec 22 2013

A320394 Number of ones in binary expansion n^5.

Original entry on oeis.org

0, 1, 1, 6, 1, 6, 6, 7, 1, 9, 6, 10, 6, 11, 7, 12, 1, 10, 9, 9, 6, 11, 10, 14, 6, 11, 11, 15, 7, 15, 12, 15, 1, 10, 10, 13, 9, 11, 9, 13, 6, 16, 11, 15, 10, 21, 14, 18, 6, 15, 11, 13, 11, 13, 15, 19, 7, 17, 15, 14, 12, 14, 15, 18, 1, 10, 10, 14, 10, 18, 13, 18

Offset: 0

Vincenzo Librandi, Nov 28 2018

The binary weight of n^5.

Cf. A000120, A000584, A118738, A192085.

Magma
```
[&+Intseq(n^5, 2): n in [0..79]];
```

Table[DigitCount[n^5, 2][[1]], {n, 0, 100}]

a(n) = hammingweight(n^5); \\ Michel Marcus, Nov 28 2018

a(n) = A000120(A000584(n)).

A364785 Primes whose binary representation has more 1-bits than its cube.

Original entry on oeis.org

445317119867, 28498383073019, 114304774692347, 7594322375176157

Offset: 1

Martin Ehrenstein, Aug 07 2023

			445317119867 is a term because it is a prime number and A000120(445317119867) = 31, while A192085(445317119867) = A000120(445317119867^3) = 30.

Chris K. Caldwell and G. L. Honaker, Jr., 445317119867, Prime Curios!

Intersection of A363799 and A000040.

Cf. A000120, A192085, A138597.

cp's OEIS Frontend

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

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A286375 a(n) = A278222(n^3).

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

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A192086 Numbers n such that the binary weight (A000120) of n^3 is less than the binary weight of n^2.

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A234218 Primes whose cubes are odious.

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A320394 Number of ones in binary expansion n^5.

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A364785 Primes whose binary representation has more 1-bits than its cube.

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