Russ Cox - cp's OEIS frontend

A249058 a(n) = number of primes less than the square root of the (2^n)-th prime.

0, 0, 1, 2, 4, 5, 7, 9, 12, 17, 24, 32, 45, 61, 82, 114, 154, 215, 293, 404, 557, 766, 1057, 1459, 2025, 2800, 3880, 5379, 7470, 10368, 14414, 20030, 27864, 38745, 53982, 75206, 104799, 146151, 203821, 284381, 396976, 554303, 774256, 1081749, 1511871, 2113506

Offset: 0

Russ Cox, Oct 19 2014

nonn

Hugo Pfoertner, Table of n, a(n) for n = 0..78 (terms 0..56 from Jens Kruse Andersen)

Cf. A033844. Related to analysis of A247665.

Mathematica
```
PrimePi[Sqrt[Prime[2^n]]]
```

a(n) = primepi(sqrt(A033844(n))). - Jens Kruse Andersen, Oct 20 2014

More terms from Jens Kruse Andersen, Oct 20 2014

A248391 Terms in A247665 which are neither primes nor prime powers, in order of appearance.

Original entry on oeis.org

15, 14, 85, 57, 161, 319, 403, 259, 451, 559, 235, 901, 177, 1159, 1541, 781, 2117, 1027, 2573, 445, 1649, 1121, 122, 707, 851, 1133, 2911, 3103, 1417, 3397, 3503, 415, 4183, 2159, 5141, 393, 2603, 244, 973, 2323, 5513, 10117, 4223, 4553, 11899, 2171, 7439, 5549, 8507, 3247, 6731, 579, 2489, 8083, 1379, 3197

Offset: 1

Russ Cox and N. J. A. Sloane, Oct 16 2014

nonn

Russ Cox and N. J. A. Sloane, Table of n, a(n) for n = 1..374

Cf. A247665, A247885, A248379, A248387-A248390.

A248390 Composite terms in A247665 in order of appearance.

Original entry on oeis.org

4, 9, 8, 15, 14, 25, 27, 16, 49, 121, 169, 85, 57, 32, 161, 319, 403, 125, 289, 81, 361, 64, 259, 529, 451, 841, 559, 961, 235, 901, 177, 1159, 128, 343, 1369, 1541, 781, 1681, 2117, 1027, 1849, 2573, 445, 2209, 1649, 2809, 243, 1121, 122, 707

Offset: 1

Russ Cox and N. J. A. Sloane, Oct 16 2014

Russ Cox and N. J. A. Sloane, Table of n, a(n) for n = 1..19456

Cf. A247665, A248388, A248391.

A248389 Terms in A247665 that are less than the previous term.

Original entry on oeis.org

8, 15, 14, 25, 16, 85, 57, 32, 161, 403, 125, 81, 64, 259, 451, 559, 235, 901, 177, 1159, 128, 343, 781, 1027, 445, 1649, 243, 1121, 122, 707, 851, 1133, 2911, 3103, 1417, 3397, 3503, 415, 4183, 2159, 5141, 393, 2603, 244, 973, 2323, 10117, 1331, 4223, 4553, 11899, 2171, 6241, 5549, 625, 7921, 3247, 6731, 579

Offset: 1

Russ Cox and N. J. A. Sloane, Oct 16 2014

nonn

All terms are composite.

Russ Cox and N. J. A. Sloane, Table of n, a(n) for n = 1..354

Cf. A247665.

A248388 Indices of composite terms in A247665.

Original entry on oeis.org

3, 6, 7, 13, 15, 27, 28, 31, 32, 40, 50, 55, 57, 63, 65, 82, 101, 111, 112, 115, 116, 127, 131, 132, 165, 172, 203, 204, 223, 225, 231, 233, 255, 263, 264, 278, 331, 332, 358, 407, 408, 417, 447, 448, 451, 455, 463, 467, 511, 527, 557, 661, 663, 665, 717, 761

Offset: 1

Russ Cox and N. J. A. Sloane, Oct 16 2014

nonn

Russ Cox and N. J. A. Sloane, Table of n, a(n) for n = 1..19456

Cf. A247665, A248387, A248379, A248390.

A248387 a(n) = index of n-th prime in A247665.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Russ Cox and N. J. A. Sloane, Oct 16 2014

nonn

Russ Cox and N. J. A. Sloane, Table of n, a(n) for n = 1..99436

Cf. A247665, A248388, A248379.

Subsequence of A248918.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a248387 = (+ 1) . fromJust . (`elemIndex` a247665_list) . a000040
-- Reinhard Zumkeller, Oct 16 2014

A248381 Multiples of 3 in A247665 in order of appearance.

Original entry on oeis.org

9, 15, 27, 57, 81, 177, 243, 393, 579, 729, 849, 1257, 2187, 4197, 4143, 5853, 6561, 17817, 17781, 19683, 56157, 59049, 104289, 104277, 154509

Offset: 1

Russ Cox and N. J. A. Sloane, Oct 12 2014

The initial multiples of 3 occur at positions 6,13,28,57,115,231, ... - conjecturally, for n > 13, A247665(n) is a multiple of 3 iff n = 29*2^k-1, k >= 0.

			Table showing index where multiples of 3 appear, the multiple of 3, and the number of different prime factors of that term:
6 9 1
13 15 2
28 27 1
57 57 2
115 81 1
231 177 2
463 243 1
927 393 2
1855 579 2
3711 729 1
7423 849 2
14847 1257 2
29695 2187 1
59391 4197 2
118783 4143 2
237567 5853 2
475135 6561 1
950271 17817 2
1900543 17781 2
3801087 19683 1
7602175 56157 2
15204351 59049 1
30408703 104289 2
60817407 104277 2
121634815 154509 2

Cf. A247665, A248379 (multiples of 2).

A178939 Maximal AND-OR-XOR formula complexity (operator count) for n-input Boolean functions.

Original entry on oeis.org

1, 1, 4, 7, 12

Offset: 1

Russ Cox, Dec 30 2010

D. E. Knuth, The Art of Computer Programming, Volume 4A, Section 7.1.2.

Cf. A056287.

A160638 Bit-reversed 8-bit binary numbers.

Original entry on oeis.org

0, 128, 64, 192, 32, 160, 96, 224, 16, 144, 80, 208, 48, 176, 112, 240, 8, 136, 72, 200, 40, 168, 104, 232, 24, 152, 88, 216, 56, 184, 120, 248, 4, 132, 68, 196, 36, 164, 100, 228, 20, 148, 84, 212, 52, 180, 116, 244, 12, 140, 76, 204, 44, 172, 108, 236, 28, 156

Offset: 0

Russ Cox, May 21 2009

This sequence is found in computer programs that need to reverse the bits in a byte, typically during data compression or other bit-level encoding. a(n) is its own inverse: a(a(n)) = n.
A permutation of the integers 0-255. - Jon Perry, Oct 06 2012
a(n) is even for 0 <= n< 128 and odd for n <= 128 < 256. - Jon Perry, Oct 06 2012
a(m) + a(n) = a(m+n) when the binary representations of m and n have no bits in common. - Jon Perry, Oct 06 2012

			n = 1 = 00000001 binary, so a(1) = 10000000 binary = 128.
n = 29 = 00011101 binary, so a(29) = 10111000 binary = 184.

Henry S. Warren, Hacker's Delight, Addison-Wesley, 2002, pages 101-106.

Russ Cox, Table of n, a(n) for n = 0 .. 255 (full sequence)
Sean Anderson, Bit Twiddling Hacks
Michael Beeler, R. William Gosper and Richard C. Schroeppel, HAKMEM (MIT AI Memo 239, Feb. 29, 1972), Item 167.

Cf. A217589.

int a = 0; for(int i=0; i<8; i++) if(n & (1<

import Data.Bits (testBit, setBit)
import Data.Word (Word8)
a160638 :: Word8 -> Word8
a160638 n = rev 0 0 where
   rev 8 y = y
   rev i y = rev (i + 1) (if testBit n i then setBit y (7 - i) else y)
-- Reinhard Zumkeller, Jan 12 2013

a:= n-> Bits[Join](ListTools[Reverse](Bits[Split](n, bits=8))):
seq(a(n), n=0..255);  # Alois P. Heinz, Nov 28 2024

a[n_] := FromDigits[PadLeft[IntegerDigits[n, 2], 8] // Reverse, 2]; Table[a[n], {n, 0, 255}] (* Jean-François Alcover, Dec 26 2015 *)
IntegerReverse[Range[0, 255], 2, 8] (* Paolo Xausa, Nov 28 2024 *)

A160638(n)=binary(n+256)*vector(9,n,2^n)~\4  \\ M. F. Hasler, Oct 07 2012

A160638(n)=sum(i=0,7,bittest(n,7-i)<M. F. Hasler, Oct 07 2012

def a(n): return int(bin(n)[2:].zfill(8)[::-1], 2)
print([a(n) for n in range(256)]) # Michael S. Branicky, Jul 13 2022

a(n) = floor(A030101(n+256)/2). - Reinhard Zumkeller, Jan 12 2013

A129760 Bitwise AND of binary representation of n-1 and n.

Original entry on oeis.org

0, 0, 2, 0, 4, 4, 6, 0, 8, 8, 10, 8, 12, 12, 14, 0, 16, 16, 18, 16, 20, 20, 22, 16, 24, 24, 26, 24, 28, 28, 30, 0, 32, 32, 34, 32, 36, 36, 38, 32, 40, 40, 42, 40, 44, 44, 46, 32, 48, 48, 50, 48, 52, 52, 54, 48, 56, 56, 58, 56, 60, 60, 62, 0, 64, 64, 66, 64, 68, 68, 70, 64, 72, 72, 74

Offset: 1

Russ Cox, May 15 2007

Also the number of Ducci sequences with period n.
Also largest number less than n having in binary representation fewer ones than n has; A048881(n-1) = A000120(a(n)) = A000120(n)-1. - Reinhard Zumkeller, Jun 30 2010
a(n) is the parent of vertex n in the binomial tree.  The binomial tree is root vertex n=0, then for n>=1 the parent of n is n with its least significant 1-bit changed to a 0-bit.  Binomial tree order 5, n=0 to 31 inclusive, is the frontispiece of Knuth volume 1, second and subsequent editions.  Vertices are shown there with n in binary dots and a(n) is the next vertex towards the root at the bottom of the page. - Kevin Ryde, Jul 24 2019

			a(6) = 6 AND 5 = binary 110 AND 101 = binary 100 = 4.

Donald E. Knuth, The Art of Computer Programming, volume 1, second edition, frontispiece. Reproduced with brief description of the art in Donald E. Knuth, Selected Papers on Fun and Games, 2010, Chapter 47 Geek Art, figure 16, page 679.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ron Brown and Jonathan L. Merzel, The number of Ducci sequences with a given period, Fib. Quart., 45 (2007), 115-121.

Cf. A038712, A086799, A104594, A059991, A006519, A109168, A220466.

C
```
int a(int n) { return n & (n-1); }
```

[n - 2^Valuation(n, 2): n in [1..100]]; // Vincenzo Librandi, Jul 25 2019

nmax := 75: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (2*n-2) * 2^p od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jun 22 2011, revised Jan 25 2013
A129760 := n -> Bits:-And(n-1, n):
seq(A129760(n), n=1..75); # Peter Luschny, Sep 26 2019

Table[BitAnd[n, n - 1], {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

a(n)=bitand(n,n-1) \\ Charles R Greathouse IV, Jun 23 2011

def a(n): return n & (n-1)
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Jul 13 2022

a(n) = n AND n-1.
Equals n - A006519(n). - N. J. A. Sloane, May 26 2008
From Johannes W. Meijer, Jun 22 2011: (Start)
a((2*n-1)*2^p) = (2*n-2)*(2^p), p>=0.
a(2*n-1) = (2*n-2), n>=1, and a(2^p+1) = 2^p, p>=1. (End)

cp's OEIS Frontend

User: Russ Cox

A249058 a(n) = number of primes less than the square root of the (2^n)-th prime.

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A248391 Terms in A247665 which are neither primes nor prime powers, in order of appearance.

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A248390 Composite terms in A247665 in order of appearance.

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A248389 Terms in A247665 that are less than the previous term.

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A248388 Indices of composite terms in A247665.

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A248387 a(n) = index of n-th prime in A247665.

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Haskell

A248381 Multiples of 3 in A247665 in order of appearance.

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A178939 Maximal AND-OR-XOR formula complexity (operator count) for n-input Boolean functions.

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A160638 Bit-reversed 8-bit binary numbers.

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A129760 Bitwise AND of binary representation of n-1 and n.

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