A070883 -id:A070883 - cp's OEIS frontend

A169810 a(n) = n XOR n^2.

0, 0, 6, 10, 20, 28, 34, 54, 72, 88, 110, 114, 156, 164, 202, 238, 272, 304, 342, 378, 388, 428, 498, 518, 600, 616, 702, 706, 780, 852, 922, 990, 1056, 1120, 1190, 1258, 1332, 1404, 1410, 1494, 1640, 1720, 1742, 1810, 1980, 1988, 2154, 2190, 2352, 2384, 2550, 2586

Offset: 0

N. J. A. Sloane, May 28 2010

XOR the binary representations of n and n^2.

			a(5) = 28:
..101 <- 5
11001 <- 25
----- <- XOR
11100 -> 28

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Suggested by A174375. Cf. A070883, A169811-A169814.

Cf. A007745 (OR), A213541 (AND), A002378.

import Data.Bits (xor)
a169810 n = n ^ 2 `xor` n :: Integer
-- Reinhard Zumkeller, Dec 27 2012

f:=proc(n) local i,t0,t1,t2,ts,tl,n1,n2;
t1:=convert(n,base,2); t2:=convert(n^2,base,2); n1:=nops(t1); n2:=nops(t2);
if n1 < n2 then ts:= t1; tl:=t2; else ts:=t2; tl:=t1; fi;
t0:=[]; for i from 1 to nops(ts) do t0:=[op(t0), (ts[i] + tl[i]) mod 2 ]; od:
for i from nops(ts)+1 to nops(tl) do t0:=[op(t0), tl[i]]; od:
add(2^(i-1)*t0[i], i=1..nops(t0)); end;
# second Maple program:
a:= n-> Bits[Xor](n, n^2):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 29 2018

a[n_]:=BitXor[n, n^2]; Array[a, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

A169810(n)=bitxor(n^2,n) \\ M. F. Hasler, May 07 2023

A169810=lambda n:n**2^n # M. F. Hasler, May 07 2023

A169813 a(n) = n XOR sigma(n), where sigma(n) is the number of divisors of n, A000203.

Original entry on oeis.org

0, 1, 7, 3, 3, 10, 15, 7, 4, 24, 7, 16, 3, 22, 23, 15, 3, 53, 7, 62, 53, 50, 15, 36, 6, 48, 51, 36, 3, 86, 63, 31, 17, 20, 19, 127, 3, 26, 31, 114, 3, 74, 7, 120, 99, 102, 31, 76, 8, 111, 123, 86, 3, 78, 127, 64, 105, 96, 7, 148, 3, 94, 87, 63, 21, 210, 7, 58, 37, 214, 15, 139, 3, 56

Offset: 1

N. J. A. Sloane, May 28 2010

Cf. A000203, A003987, A070883, A169810, A169811, A169812, A169814.

Cf. also A106409, A178910, A227320, A294899.

a(n) = bitxor(n, sigma(n)); \\ Michel Marcus, Nov 25 2017

(define (A169813 n) (A003987bi n (A000203 n))) ;; Where A003987bi implements the bitwise-XOR, A003987 and code for A000203 can be found under that entry. - Antti Karttunen, Nov 25 2017

A169811 a(n) = n XOR n*(n+1)/2.

Original entry on oeis.org

0, 0, 1, 5, 14, 10, 19, 27, 44, 36, 61, 73, 66, 86, 103, 119, 152, 136, 185, 173, 198, 242, 235, 259, 308, 348, 325, 353, 394, 430, 463, 495, 560, 528, 625, 597, 702, 666, 707, 811, 796, 884, 941, 921, 1010, 1062, 1047, 1095, 1192, 1272, 1225, 1309, 1366, 1442, 1531

Offset: 0

N. J. A. Sloane, May 28 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000217, A070883, A169810.

Table[BitXor[n (n + 1)/2, n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011 *)

a(n)=bitxor(n, n*(n+1)/2); \\ Michel Marcus, Jul 08 2019

A169814 a(n) = n XOR phi(n).

Original entry on oeis.org

0, 3, 1, 6, 1, 4, 1, 12, 15, 14, 1, 8, 1, 8, 7, 24, 1, 20, 1, 28, 25, 28, 1, 16, 13, 22, 9, 16, 1, 22, 1, 48, 53, 50, 59, 40, 1, 52, 63, 56, 1, 38, 1, 56, 53, 56, 1, 32, 27, 38, 19, 44, 1, 36, 31, 32, 29, 38, 1, 44, 1, 32, 27, 96, 113, 86, 1, 100, 105, 94, 1, 80, 1, 110, 99, 104, 113, 86

Offset: 1

N. J. A. Sloane, May 28 2010

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

Cf. A000010, A065091, A070883, A169810, A169811, A169812, A169813.

a:= n-> Bits[Xor](n, numtheory[phi](n)):
seq(a(n), n=1..89);  # Alois P. Heinz, Jul 06 2023

Table[BitXor[n,EulerPhi[n]],{n,80}] (* Harvey P. Dale, Sep 18 2011 *)

a(n)=bitxor(n,eulerphi(n)) \\ Charles R Greathouse IV, Feb 21 2013

a(n) = 1 <=> n in { A065091 }. - Alois P. Heinz, Jul 06 2023

A169812 a(n) = n XOR d(n) (cf. A000005).

Original entry on oeis.org

0, 0, 1, 7, 7, 2, 5, 12, 10, 14, 9, 10, 15, 10, 11, 21, 19, 20, 17, 18, 17, 18, 21, 16, 26, 30, 31, 26, 31, 22, 29, 38, 37, 38, 39, 45, 39, 34, 35, 32, 43, 34, 41, 42, 43, 42, 45, 58, 50, 52, 55, 50, 55, 62, 51, 48, 61, 62, 57, 48, 63, 58, 57, 71, 69, 74, 65, 66, 65, 78, 69, 68, 75, 78

Offset: 1

N. J. A. Sloane, May 28 2010

nonn

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

Cf. A000005, A070883, A169810, A169811, A169813, A169814.

a[n_] := BitXor[n, DivisorSigma[0, n]]; Array[a, 100] (* Amiram Eldar, Jul 08 2019 *)

a(n)=bitxor(n, numdiv(n)); \\ Michel Marcus, Jul 08 2019

A265885 a(n) = n IMPL prime(n), where IMPL is the bitwise logical implication.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 25, 23, 23, 29, 31, 55, 59, 59, 63, 63, 63, 61, 111, 111, 107, 111, 123, 127, 103, 101, 103, 107, 111, 113, 127, 223, 223, 223, 221, 223, 223, 251, 255, 255, 247, 245, 255, 211, 215, 215, 211, 223, 239, 237, 237, 239, 251, 251, 457, 455

Offset: 1

Reinhard Zumkeller, Dec 17 2015

nonn

			.   prime(25)=97 | 1100001
.             25 |   11001
.   -------------+--------
.     25 IMPL 97 | 1100111 -> a(25) = 103 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Implies

Cf. A000040, A004676, A007088, A070883 (XOR), A265705.

a265885 n = n `bimpl` a000040 n where
   bimpl 0 0 = 0
   bimpl p q = 2 * bimpl p' q' + if u <= v then 1 else 0
               where (p', u) = divMod p 2; (q', v) = divMod q 2

using IntegerSequences
[Bits("IMP", n, p) for (n, p) in enumerate(Primes(1, 263))] |> println  # Peter Luschny, Sep 25 2021

a:= n-> Bits[Implies](n, ithprime(n)):
seq(a(n), n=1..56);  # Alois P. Heinz, Sep 24 2021

IMPL[n_, k_] := If[n == 0, 0, BitOr[2^Length[IntegerDigits[k, 2]]-1-n, k]];
a[n_] := n ~IMPL~ Prime[n];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 25 2021, after David A. Corneth's code in A265705 *)

a(n) = bitor((2<Michel Marcus, Jan 22 2022

a(n) = A265705(A000040(n),n).

A194187 Difference of n-th prime and (bitwise XOR of n and n-th prime).

Original entry on oeis.org

-1, 2, -1, 4, -3, 2, -5, -8, -7, 6, 11, -4, 5, 6, 15, 16, 17, 14, -13, -12, -19, -10, 15, 24, -23, -26, -21, -12, -3, 2, 31, -32, -31, -30, -33, -28, -27, 30, 39, 40, 25, 22, 43, -44, -35, -34, -41, -16, 17, 14, 15, 20, 45, 46, -53, -56, -39, -38, -25, -12

Offset: 1

Reinhard Zumkeller, Oct 12 2011

a(n) = A000040(n) - A070883(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A070883, A174375.

a194187_list = zipWith (-) a000040_list a070883_list

#[[2]]-BitXor[#[[1]],#[[2]]]&/@Table[{n,Prime[n]},{n,60}] (* Harvey P. Dale, Oct 05 2023 *)

A344856 Bitwise XOR of prime(n) and n^2.

Original entry on oeis.org

3, 7, 12, 23, 18, 41, 32, 83, 70, 121, 102, 181, 128, 239, 206, 309, 282, 377, 298, 471, 496, 427, 578, 537, 528, 705, 702, 891, 804, 1013, 958, 1155, 1224, 1039, 1116, 1415, 1476, 1287, 1366, 1773, 1570, 1617, 1926, 1873, 1836, 2179, 2162, 2527, 2434, 2337

Offset: 1

Chris von Csefalvay, May 30 2021

This is effectively the bitwise XOR of A000040 and A000290.

			For n=3, a(3) is prime(3) XOR 3^2 = 5 XOR 9 or b(0101) XOR b(1001) = (b)1100, which in base 10 is 12.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Chris von Csefalvay, Bitwise prime-square sequences ("Jellyfish Hearts").
Wikipedia, Bitwise operation

Cf. A000040, A000290, A003987, A070883, A169810.

a:= n-> Bits[Xor](n^2, ithprime(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, May 30 2021

a[n_] := BitXor[n^2, Prime[n]]; Array[a, 50] (* Amiram Eldar, Jun 05 2021 *)

A344856(n) = bitxor(prime(n),n*n); \\ Antti Karttunen, Jun 05 2021

from sympy import primerange, prime
import numpy
def a_vector(n):
    primes = list(primerange(0, prime(n)))
    squares = [x ** 2 for x in range(1, n)]
    return numpy.bitwise_xor(primes, squares)

from sympy import prime
def A344856(n): return prime(n) ^ n**2 # Chai Wah Wu, Jun 12 2021

a(n) = prime(n) XOR n^2.

a(n) = A003987(A000040(n), A000290(n)).

cp's OEIS Frontend

A169810 a(n) = n XOR n^2.

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A169813 a(n) = n XOR sigma(n), where sigma(n) is the number of divisors of n, A000203.

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A169811 a(n) = n XOR n*(n+1)/2.

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A169814 a(n) = n XOR phi(n).

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A169812 a(n) = n XOR d(n) (cf. A000005).

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A265885 a(n) = n IMPL prime(n), where IMPL is the bitwise logical implication.

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A194187 Difference of n-th prime and (bitwise XOR of n and n-th prime).

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A344856 Bitwise XOR of prime(n) and n^2.

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