A112591 -id:A112591 - cp's OEIS frontend

A375235 Records of A112591.

1, 6, 12, 28, 58, 126, 252, 506, 1012, 2042, 4082, 8190, 16366, 32742, 65518, 131056, 262114, 524280, 1048554, 2097146, 4194278, 8388594, 16777208, 33554390, 67108858, 134217716, 268435396, 536870852, 1073741814, 2147483614, 4294967284, 8589934580, 17179869158

Offset: 1

Bill McEachen, Aug 06 2024

nonn

Sequence closely parallel to A000295.

			The first term of A112591 = 1 is a record and is a(1). The next A112591 value > 1 is 6 which is a(2).

Cf. A000295, A014210 (primes where records occur), A014234, A112591.

a[n_] := BitXor @@ NextPrime[2^n, {-1, 1}]; a[1] = 1; Array[a, 33] (* Amiram Eldar, Aug 08 2024 *)

a(n)= if(n==1,1,bitxor(precprime(2^n), nextprime(2^n) ))

a(n) = previous_prime(2^n) XOR next_prime(2^n) = A112591(A014234(n)) for n > 1.

More terms from Amiram Eldar, Aug 06 2024

A126084 a(n) = XOR of first n primes.

Original entry on oeis.org

0, 2, 1, 4, 3, 8, 5, 20, 7, 16, 13, 18, 55, 30, 53, 26, 47, 20, 41, 106, 45, 100, 43, 120, 33, 64, 37, 66, 41, 68, 53, 74, 201, 64, 203, 94, 201, 84, 247, 80, 253, 78, 251, 68, 133, 64, 135, 84, 139, 104, 141, 100, 139, 122, 129, 384, 135, 394, 133, 400, 137, 402, 183, 388, 179

Offset: 0

Esko Ranta, Mar 02 2007

The values at odd positive indices are even and the values at even positive indices are odd.
Does this sequence contain any zeros for n > 0? Probabilistically, one would expect so; but none in first 10000 terms. - Franklin T. Adams-Watters, Jul 17 2011
None below 1.5 * 10^11: any prime p such that a(pi(p)) = 0 is 43 bits or longer. Heuristic chances that a prime below 2^100 yields 0 are about 45%. Note that an n-bit prime can yield 0 only if a(pi(p)) is odd, where p is the smallest n-bit prime. That is, for n > 1, there are no zeros from pi(2^n) to pi(2^(n+1)) if A007053(n) is even. - Charles R Greathouse IV, Jul 17 2011

			a(4) = 3 because ((2 XOR 3) XOR 5) XOR 7 = (1 XOR 5) XOR 7 = 4 XOR 7 = 3
[Or, in base 2]
((10 XOR 11) XOR 101) XOR 111 = (1 XOR 101) XOR 111 = 100 XOR 111 = 11

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000

Cf. A003815, A004767, A112591.

Module[{nn=70,prs},prs=Prime[Range[nn]];Table[BitXor@@Take[prs,n],{n,0,nn}]] (* Harvey P. Dale, Jun 23 2016 *)

al(n)=local(m);vector(n,k,m=bitxor(m,prime(k))) /* Produces a vector without a(0) = 0; Franklin T. Adams-Watters, Jul 17 2011 */

v=primes(300); for(i=2,#v,v[i]=bitxor(v[i],v[i-1])); concat(0, v) \\ Charles R Greathouse IV, Aug 26 2014

q=0; forprime(p=2, 313, print1(q, ","); q=bitxor(q, p)) /* Klaus Brockhaus, Mar 06 2007; adapted by Rémy Sigrist, Oct 23 2017 */

from operator import xor
from functools import reduce
from sympy import primerange, prime
def A126084(n): return reduce(xor,primerange(2,prime(n)+1)) if n else 0 # Chai Wah Wu, Jul 09 2022

a(0) = 0; a(n) = a(n-1) XOR prime(n).

More terms from Klaus Brockhaus, Mar 06 2007
Edited by N. J. A. Sloane, Oct 22 2017 (merging old entry A193174 with this)
Edited by Rémy Sigrist, Oct 23 2017

A283750 a(n) = n^2 XOR (n + 1)^2.

Original entry on oeis.org

1, 5, 13, 25, 9, 61, 21, 113, 17, 53, 29, 233, 57, 109, 37, 481, 33, 101, 45, 249, 41, 93, 1013, 81, 49, 213, 125, 457, 89, 205, 69, 1985, 65, 197, 77, 473, 73, 253, 85, 945, 209, 117, 477, 169, 121, 4013, 229, 417, 97, 165, 1005, 185, 105, 413, 181, 1937, 241, 405, 189, 905, 153, 397, 133, 8065, 129, 389, 141, 921

Offset: 0

Ilya Gutkovskiy, Mar 15 2017

nonn

XOR the binary representations of n^2 and (n + 1)^2.

Antti Karttunen, Table of n, a(n) for n = 0..20000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, XOR
Index entries for sequences related to binary expansion of n

Cf. A000290, A003987, A016813 (records), A038712, A112591, A169810.

Cf. also A379007.

Table[BitXor[n^2, (n + 1)^2], {n, 0, 67}]

A283750(n) = bitxor(n^2, (n+1)^2); \\ Indranil Ghosh, Mar 15 2017, Antti Karttunen, Dec 16 2024

def A283750(n): return (n**2)^(n + 1)**2 # Indranil Ghosh, Mar 15 2017

a(n) = A000290(n) XOR A000290(n+1).

A334143 a(n) = bitwise NOR of prime(n) and prime(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 12, 8, 0, 0, 0, 18, 20, 16, 0, 0, 0, 0, 56, 48, 48, 32, 36, 6, 26, 24, 16, 16, 2, 0, 0, 116, 116, 96, 104, 96, 64, 88, 80, 64, 72, 64, 0, 58, 56, 40, 32, 0, 24, 18, 16, 0, 4, 4, 248, 240, 240, 224, 226, 228, 192, 200, 200, 192, 194, 128, 164

Offset: 1

Christoph Schreier, Apr 15 2020

			a(6) = prime(6) NOR prime(7) = 13 NOR 17 = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Wikipedia, Bitwise operation

Cf. A000040, A112591, A145382, A175330.

a:= n-> Bits[Nor](ithprime(n), ithprime(n+1)):
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 15 2020

A334143[n_]:=With[{b=BitOr[Prime[n],Prime[n+1]]},2^BitLength[b]-b-1];Array[A334143,100] (* Paolo Xausa, Oct 13 2023 *)

a(n) = my(x=bitor(prime(n), prime(n+1))); bitneg(x, #binary(x)); \\ Michel Marcus, Apr 16 2020

def NORprime(n):
    s = str(bin(primes[n]))[2:]
    t = str(bin(primes[n-1]))[2:]
    k = (len(s) -  len(t))
    t = k*'0' + t
    r = ''
    for i in range(len(s)):
        if s[i] == t[i] and s[i] == '0':
            r += '1'
        else:
            r += '0'
    return int(r,2)

a(n) = A035327(A175329(n)).

A334172 Bitwise XNOR of prime(n) and prime(n + 1).

Original entry on oeis.org

2, 1, 5, 3, 9, 3, 29, 27, 21, 29, 5, 51, 61, 59, 37, 49, 57, 1, 123, 113, 121, 99, 117, 71, 123, 125, 115, 121, 99, 113, 3, 245, 253, 225, 253, 245, 193, 251, 245, 225, 249, 245, 129, 251, 253, 235, 243, 195, 249, 243, 249, 225, 245, 5, 505, 501, 509, 485

Offset: 1

Christoph Schreier, Apr 17 2020

XOR is exclusive OR, meaning that one bit is on and the other bit is off. XNOR is the negation of XOR, meaning that either both bits are on or both bits are off. For example, 4 in binary is 100 and 6 is 110. Then 100 XOR 110 is 010 but 100 XNOR 110 is 101.

From Bertrand's postulate it follows that prime(n + 1) requires only one bit more than prime(n) if they're not the same bit width. In most computer implementations, however, the numbers are placed zero-padded into fixed bit widths using two's complement, making it necessary to make adjustments to avoid unintentionally negative numbers. - Alonso del Arte, Apr 18 2020

			The second prime is 3 (11 in binary) and the third prime is 5 (101 in binary). We see that 011 XNOR 101 = 001. Hence a(2) = 1.
The fourth prime is 7 (111 in binary). We see that 101 XNOR 111 = 101. Hence a(3) = 5.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, XNOR
Wikipedia, Bitwise operation

Cf. A000040, A112591, A175329, A175330, A334143.

a:= n-> (p-> Bits[Not](Bits[Xor](p, ithprime(n+1)),
             bits=1+ilog2(p)))(ithprime(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 17 2020

Table[BitNot[BitXor[Prime[n], Prime[n + 1]]] + 2^Ceiling[Log[2, Prime[n + 1]]], {n, 50}] (* Alonso del Arte, Apr 17 2020 *)

neg(p) = bitneg(p, #binary(p));
a(n) = my(p=prime(n), q=nextprime(p+1)); bitor(bitand(p, q), bitand(neg(p), neg(q))); \\ Michel Marcus, Apr 17 2020

def XNORprime(n):
    return ~(primes[n] ^ primes[n+1]) + (1 << primes[n+1].bit_length())

val prime: LazyList[Int] = 2 #:: LazyList.from(3).filter(i => prime.takeWhile {
   j => j * j <= i
}.forall {
   k => i % k != 0
})
(0 to 63).map(n => ~(prime(n) ^ prime(n + 1)) + 2 * Integer.highestOneBit(prime(n + 1))) // Alonso del Arte, Apr 18 2020

a(n) = A035327(A112591(n)).

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A375235 Records of A112591.

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A126084 a(n) = XOR of first n primes.

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A283750 a(n) = n^2 XOR (n + 1)^2.

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A334143 a(n) = bitwise NOR of prime(n) and prime(n+1).

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A334172 Bitwise XNOR of prime(n) and prime(n + 1).

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