A017912 -id:A017912 - cp's OEIS frontend

A017910 Powers of sqrt(2) rounded down.

1, 1, 2, 2, 4, 5, 8, 11, 16, 22, 32, 45, 64, 90, 128, 181, 256, 362, 512, 724, 1024, 1448, 2048, 2896, 4096, 5792, 8192, 11585, 16384, 23170, 32768, 46340, 65536, 92681, 131072, 185363, 262144, 370727, 524288

Offset: 0

N. J. A. Sloane

a(n) is the number of positive squares <= 2^n (cf. A136417). - Hans Havermann, Apr 05 2008
If expressed to two significant digits, these are the f-stop numbers in photography: 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, ...
There are also "half stops" (sqrt(2)^(n/2)) and "third stops" (sqrt(2)^(n/3)): 1, 1.4, 1.6, 1.8, 2.0, 2.2, 2.5, 2.8, 3.2, 3.6, 4, 4.5, 5, 5.7, 6.3, 7.1, 8, 9, 10.
a(n) is also the ratio (rounded down) of the curvature of the circle inscribed in the n-th 45-45-90 triangle to that of the circle inscribed in the 1st triangle, with the triangles arranged in a spiral as shown in the illustration in the links section. - Kival Ngaokrajang, Aug 28 2013
a(n) is also the total length of Heighway dragon (rounded down) after n-iterations when L(0) = 1. See illustration in links. - Kival Ngaokrajang, Dec 15 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms.
Kival Ngaokrajang, Illustration of Heighway dragon for n = 0..5.
Wikipedia, Dragon curve.

Cf. A136417, A017912. Bisections: A000079, A084188.

Partial sums of A190568.

[Floor(Sqrt(2^n)): n in [0..40]]; // Vincenzo Librandi, Nov 20 2011

[Isqrt(2^n):n in[0..40]]; // Jason Kimberley, Oct 25 2016

A017910 := n->floor(sqrt(2^n)); # Peter Luschny, Sep 20 2011

Floor[(Sqrt[2])^Range[0,40]] (* Vincenzo Librandi, Nov 20 2011 *)

a(n)=sqrtint(2^n) \\ Charles R Greathouse IV, Sep 22 2011

from math import isqrt
def A017910(n): return isqrt(1<>1) # Chai Wah Wu, Jul 26 2022

a(n) = A000196(A000079(n)). - Jason Kimberley, Oct 28 2016

a(n) = A017912(n)-1 if n is odd. a(n) = A017912(n) = 2^(n/2) if n is even. - Chai Wah Wu, Jul 26 2022

A329202 a(n) = floor(2*log_2(n)) = floor(log_2(n^2)).

Original entry on oeis.org

0, 2, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

M. F. Hasler, Nov 07 2019

Cf. A017912, A328911 (has this as row lengths).

PARI
```
a(n)=logint(n^2,2)
```

A357753 a(n) is the least square with n binary digits.

Original entry on oeis.org

4, 9, 16, 36, 64, 144, 256, 529, 1024, 2116, 4096, 8281, 16384, 33124, 65536, 131769, 262144, 525625, 1048576, 2099601, 4194304, 8392609, 16777216, 33558849, 67108864, 134235396, 268435456, 536895241, 1073741824, 2147488281, 4294967296, 8589953124, 17179869184

Offset: 3

Hugo Pfoertner, Oct 11 2022

Cf. A000290, A000302, A065732, A070939, A017912, A357754.

a:= n-> ceil(sqrt(2)^(n-1))^2:
seq(a(n), n=3..35);  # Alois P. Heinz, Oct 13 2022

Array[Ceiling[Sqrt[2^(# - 1)]]^2 &, 33, 3]

for (n=3, 35, for(k=0, oo, if(#digits(k^2,2)==n, print1(k^2,", "); break)))

a(n) = if(n%2 == 1, 1 << (n-1), ceil(sqrt(1<<(n-1)))^2) \\ David A. Corneth, Oct 11 2022

from math import isqrt
def A357753(n): return 1<Chai Wah Wu, Oct 13 2022

a(n) = A017912(n-1)^2.

a(2n+1) = (2^n)^2 = 4^n, for n>=1; indeed: 4^n_{10} = 10^(2n){2} that is the least number with 2n+1 binary digits. - _Bernard Schott, Oct 15 2022

A056007 Difference between 2^n and largest square strictly less than 2^n.

Original entry on oeis.org

1, 1, 3, 4, 7, 7, 15, 7, 31, 28, 63, 23, 127, 92, 255, 7, 511, 28, 1023, 112, 2047, 448, 4095, 1792, 8191, 7168, 16383, 5503, 32767, 22012, 65535, 88048, 131071, 166831, 262143, 296599, 524287, 444943, 1048575, 296863, 2097151, 1187452, 4194303

Offset: 0

Henry Bottomley, Jul 24 2000

Note that this is not a strictly ascending sequence. - Alonso del Arte, Apr 28 2022

			a(5) = 2^5 - 5^2 =  7;
a(6) = 2^6 - 7^2 = 15.

Cf. A000079, A017912, A071797, A357754.

Cf. A051213, A056008.

Table[2^n - Floor[Sqrt[2^n - Boole[EvenQ[n]]]]^2, {n, 0, 47}] (* Alonso del Arte, Apr 28 2022 *)

a(n) = if(n%2, sqrtint(1<Kevin Ryde, Oct 12 2022

from math import isqrt
def a(n): return 2**n - isqrt(2**n-1)**2
print([a(n) for n in range(43)]) # Michael S. Branicky, Apr 29 2022

a(n) = 2^n - (ceiling(2^(n/2)) - 1)^2 = A000079(n) - (A017912(n) - 1)^2. - Vladeta Jovovic, May 01 2003
a(n) = A071797(A000079(n)). - Michel Marcus, Apr 29 2022
a(n) = 2^n - A357754(n). - Kevin Ryde, Oct 12 2022

A365931 a(n) = number of pairs {x,y} with (x,y > 1) such that x^y (= terms of A072103) has bit length <= n.

Original entry on oeis.org

0, 0, 1, 3, 7, 10, 18, 25, 35, 50, 69, 94, 132, 178, 244, 334, 460, 629, 869, 1201, 1668, 2314, 3223, 4493, 6280, 8793, 12322, 17288, 24286, 34139, 48036, 67630, 95274, 134285, 189349, 267090, 376880, 531942, 750991, 1060463, 1497741, 2115669, 2988957, 4223225, 5967822, 8433889

Offset: 1

Karl-Heinz Hofmann, Oct 07 2023

Number of pairs {x,y} with (x,y > 1) for which x^y < 2^n-1.
In some special cases different pairs have the same result (see A072103 and the example here) and those multiple representations are counted separately.
There is no need to include 2^n-1 because it is a Mersenne number and it cannot be a power anyway.
Limit_{n->oo} a(n)/a(n-1) = sqrt(2) = A002193.
Partial sums of A365930.

			For n = 6: the Mersenne number 2^6-1 = 63 is the largest number with bit length 6 and the upper bound for the following a(6) = 10 powers: 2^2, 2^3, 2^4, 2^5, 3^2, 3^3, 4^2, 5^2, 6^2, 7^2.

Cf. A072103, A002193, A365930 (first differences).

Cf. A017912 (squares), A017981 (cubes).

a[n_] := Sum[Ceiling[2^(n/k)] - 2, {k, 2, n}]; Array[a, 47]

from sympy import integer_nthroot, integer_log
def A365931(n):
    result, nMersenne, new = 0, (1<

a(n) = Sum_{y = 2..n} (ceiling(2^(n/y)) - 2)
a(n) = Sum_{y = 2..n} (floor((2^n-1)^(1/y)) - 1)
a(n) = Sum_{k = 1..n} A365930(k).

A017930 Powers of sqrt(8) rounded up.

Original entry on oeis.org

1, 3, 8, 23, 64, 182, 512, 1449, 4096, 11586, 32768, 92682, 262144, 741456, 2097152, 5931642, 16777216, 47453133, 134217728, 379625063, 1073741824, 3037000500, 8589934592, 24296004000, 68719476736

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

[Ceiling(Sqrt(8^n)): n in [0..30]]; // Vincenzo Librandi, Nov 21 2011

Ceiling[(Sqrt[8])^Range[0,40]] (* Vincenzo Librandi, Nov 21 2011 *)

a(n)=ceil(sqrt(8)^n) \\ Charles R Greathouse IV, Nov 21 2011

a(n) = A017912(3n). - R. J. Mathar, Apr 28 2008

cp's OEIS Frontend

A017910 Powers of sqrt(2) rounded down.

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A329202 a(n) = floor(2*log_2(n)) = floor(log_2(n^2)).

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A357753 a(n) is the least square with n binary digits.

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A056007 Difference between 2^n and largest square strictly less than 2^n.

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A365931 a(n) = number of pairs {x,y} with (x,y > 1) such that x^y (= terms of A072103) has bit length <= n.

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A017930 Powers of sqrt(8) rounded up.

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