A175330 -id:A175330 - cp's OEIS frontend

A175329 a(n) = bitwise OR of prime(n) and prime(n+1).

3, 7, 7, 15, 15, 29, 19, 23, 31, 31, 63, 45, 43, 47, 63, 63, 63, 127, 71, 79, 79, 95, 91, 121, 101, 103, 111, 111, 125, 127, 255, 139, 139, 159, 151, 159, 191, 167, 175, 191, 183, 191, 255, 197, 199, 215, 223, 255, 231, 237, 239, 255, 251, 507, 263, 271, 271, 287

Offset: 1

Leroy Quet, Apr 07 2010

Read each binary representation of the primes from right to left and then OR respective digits to form the binary equivalent of each term of this sequence.

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

Cf. A000040, A175330 (bitwise AND).

Cf. A086799 (bitwise OR of n and n-1).

read("transforms") ; A175329 := proc(n) ORnos(ithprime(n),ithprime(n+1)) ; end proc: seq(A175329(n),n=1..60) ; # R. J. Mathar, Apr 15 2010
# second Maple program:
a:= n-> Bits[Or](ithprime(n), ithprime(n+1)):
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 16 2020

A175329[n_]:=BitOr[Prime[n],Prime[n+1]];Array[A175329,100] (* Paolo Xausa, Oct 13 2023 *)
BitOr@@#&/@Partition[Prime[Range[60]],2,1] (* Harvey P. Dale, Mar 01 2024 *)

a(n) = bitor(prime(n), prime(n+1)); \\ Michel Marcus, Apr 20 2020

More terms from R. J. Mathar, Apr 15 2010

A214415 Numbers n such that prevprime(2^n) AND nextprime(2^n) = 1, where AND is the bitwise AND operator.

Original entry on oeis.org

2, 4, 6, 8, 12, 15, 16, 23, 25, 30, 37, 53, 55, 57, 67, 75, 76, 81, 82, 84, 95, 108, 129, 132, 135, 139, 143, 155, 160, 163, 180, 181, 188, 192, 203, 204, 210, 222, 244, 263, 273, 277, 280, 287, 289, 295, 297, 308, 315, 319, 325, 330, 341, 367, 370, 393, 394, 406

Offset: 0

Alex Ratushnyak, Aug 07 2012

A007053(a(n)) are indices of 1's in A175330. That is, A175330(A007053(a(n)))=1.

Conjecture: the sequence is infinite.

			4 is in the sequence because (prevprime(2^4) AND nextprime(2^4)) = 13 AND 17 = 1.

Cf. A007053, A175330.

import java.math.BigInteger;
public class A214415 {
  public static void main (String[] args) {
    BigInteger b1 = BigInteger.valueOf(1);
    BigInteger b2 = BigInteger.valueOf(2);
    for (int n=2; ; n++) {
      BigInteger pwr = b1.shiftLeft(n);
      BigInteger pm  = pwr.subtract(b1);
      BigInteger pp  = pwr.add(b1);
      while (true) {
        if (pm.isProbablePrime(2)) {
            if (pm.isProbablePrime(80)) break;
        }
        pm  = pm.subtract(b2);
      }
      while (true) {
        if (pp.isProbablePrime(2)) {
            if (pp.isProbablePrime(80)) break;
        }
        pp  = pp.add(b2);
      }
      if (pm.and(pp).equals(b1)) {
        System.out.printf("%d, ",n);
      }
    }
  }
}

ba1Q[n_]:=Module[{c=2^n},BitAnd[NextPrime[c],NextPrime[c,-1]]==1]; Select[ Range[ 450],ba1Q] (* Harvey P. Dale, Dec 25 2012 *)

{ for (n=2,1000,  N = 2^n;
    p1 = precprime(N-1);
    p2 = nextprime(N+1);
    ba = bitand(p1, p2);
    if ( bitand( ba, ba-1 ) == 0, print1(n,", "));
); }
/* Joerg Arndt, Aug 16 2012 */

from itertools import islice
from sympy import prevprime, nextprime
def A214415_gen(): # generator of terms
    n, m = 2, 4
    while True:
        if prevprime(m)&nextprime(m) == 1:
            yield n
        n += 1
        m *= 2
A214415_list = list(islice(A214415_gen(),20)) # Chai Wah Wu, Oct 16 2023

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)).

A328266 a(n) is the least k > 0 such that prime(n) AND prime(n+k) <= 1 (where prime(n) denotes the n-th prime number and AND denotes the bitwise AND operator).

Original entry on oeis.org

2, 1, 2, 3, 2, 1, 5, 4, 4, 9, 14, 7, 6, 21, 29, 3, 27, 1, 14, 13, 11, 33, 10, 8, 7, 6, 6, 7, 3, 2, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 43, 42, 44, 48, 39, 41, 45, 36, 35, 34, 41, 40, 49, 30, 47, 31, 27, 26, 43

Offset: 1

Rémy Sigrist, Oct 16 2019

The sequence is well defined: for any n > 0, let x be such that prime(n) < 2^x; as 1 and 2^x are coprime, by Dirichlet's theorem on arithmetic progressions, there is a prime number q of the form q = 1 + k * 2^x, and prime(n) AND q <= 1, QED.

a(n) >= A000720(A062383(A000040(n)))+1-n. - Robert Israel, Oct 17 2019

			For n = 18:
- prime(18) = 61,
- prime(19) = 67,
- 61 AND 67 = 1,
- so a(18) = 1.

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

Cf. A000040, A000720, A062383, A175330, A214415.

f:= proc(n) local L,M,R,j,v,i,x;
  L:= convert(ithprime(n),base,2);
  v:= 2^nops(L);
  M:= select(t -> L[t]=0, [$2..nops(L)]);
  for i from 1 do
    for j from 0 to 2^nops(M)-1  do
      R:= convert(j,base,2);
      x:= 1 + add(2^(M[i]-1), i=select(k -> R[k]=1, [$1..nops(R)]))+i*v;
      if isprime(x) then return numtheory:-pi(x)-n fi
  od od;
end proc:
map(f, [$1..100]); # Robert Israel, Oct 17 2019

A328266[n_]:=Module[{q=n,p=Prime[n]},While[BitAnd[p,Prime[++q]]>1];q-n];Array[A328266,100] (* Paolo Xausa, Oct 13 2023 *)

{ forprime (p=2, prime(73), k=0; forprime (q=p+1, oo, k++; if (bitand(p, q)<=1, print1 (k ", "); break))) }

a(n) = 1 iff A175330(n) = 1.

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)).

A366550 Numbers k such that bitwise AND of prime(k) and prime(k+1) = 1.

Original entry on oeis.org

2, 6, 18, 54, 564, 3512, 6542, 564163, 2063689, 54400028, 5586502348, 252252704148404, 971269945245201, 3745011184713964

Offset: 1

Paolo Xausa, Oct 13 2023

Suggested by a comment by Alex Ratushnyak in A175330.

			18 is a term since prime(18) AND prime(19) = 1,
  prime(18) = 61 = binary 0111101
  prime(19) = 67 = binary 1000011
  bitwise AND    =        0000001

Positions of ones in A175330.

Cf. A007053, A214415.

A366550list[upto_]:=PrimePi[Select[2^Range[upto],BitAnd[NextPrime[#],NextPrime[#,-1]]==1&]];
A366550list[37] (* Uses formula, considering values in A214415 up to 37 *)

isok(k) = bitand(prime(k), prime(k+1)) == 1; \\ Michel Marcus, Oct 14 2023

a(n) = A007053(A214415(n-1)).

A334150 Primes p such that p AND q = 1, where q is the next prime after p and AND is the bitwise operation.

Original entry on oeis.org

3, 13, 61, 251, 4093, 32749, 65521, 8388593, 33554393, 1073741789, 137438953447, 9007199254740881, 36028797018963913, 144115188075855859, 147573952589676412909, 37778931862957161709471, 75557863725914323419121, 2417851639229258349412301, 4835703278458516698824647

Offset: 1

Michel Marcus, Apr 16 2020

Cf. A175330.
Subsequence of A014234 (largest prime <= 2^n).
Cf. A214415 (exponents of corresponding powers of 2).

s = {}; p = 2; Do[q = NextPrime[p]; If[BitAnd[p, q] == 1, AppendTo[s, p]]; p = q, {10^5}]; s (* Amiram Eldar, Apr 16 2020 *)
Select[ NextPrime[ 2^Range[82], -1], BitAnd[#, NextPrime@ #] == 1 &] (* Giovanni Resta, Apr 16 2020 *)

isok(p) = isprime(p) && (bitand(p, nextprime(p+1)) == 1);

a(9)-a(10) from Amiram Eldar, Apr 16 2020

a(11)-a(19) from Giovanni Resta, Apr 16 2020

cp's OEIS Frontend

A175329 a(n) = bitwise OR of prime(n) and prime(n+1).

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A214415 Numbers n such that prevprime(2^n) AND nextprime(2^n) = 1, where AND is the bitwise AND operator.

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

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A328266 a(n) is the least k > 0 such that prime(n) AND prime(n+k) <= 1 (where prime(n) denotes the n-th prime number and AND denotes the bitwise AND operator).

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

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A366550 Numbers k such that bitwise AND of prime(k) and prime(k+1) = 1.

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A334150 Primes p such that p AND q = 1, where q is the next prime after p and AND is the bitwise operation.

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