A205510 Binary Hamming distance between prime(n) and prime(n+1).

1, 2, 1, 2, 2, 3, 1, 1, 2, 1, 4, 2, 1, 1, 3, 3, 2, 6, 1, 3, 2, 3, 2, 3, 1, 1, 2, 2, 3, 3, 6, 2, 1, 4, 1, 2, 5, 1, 2, 4, 2, 2, 6, 1, 1, 2, 2, 4, 2, 2, 2, 4, 2, 7, 2, 2, 1, 3, 2, 1, 5, 3, 1, 3, 1, 5, 3, 2, 2, 4, 2, 1, 3, 3, 1, 6, 1, 3, 1, 4, 2, 2, 4, 2, 2, 5, 1, 1, 1, 3, 2, 3, 2, 2, 1, 2, 7, 1, 3, 5

Offset: 1

Author

Vladimir Shevelev, Jan 28 2012

Keywords

• nonn
• base
• easy

Comments

We call "Hamming's twin primes" the pairs of consecutive primes (p,q) with Hamming distance 1. They are (2,3), (5,7), (17,19,), (19,23), (29,31), (41,43), (43,47), (67,71), (97,101), ..., (A205511,A205302). As in Twin Primes Conjecture, we conjecture that there exist infinitely many Hamming's twin pairs.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A205511, A205302, A205509, A001511, A345985.

Programs

Maple

a:= n-> add(i, i=Bits[GetBits](Bits[Xor](ithprime(n), ithprime(n+1)), 0..-1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 11 2017

Mathematica

Table[Count[IntegerDigits[BitXor[Prime[n],Prime[n+1]],2],1],{n,100}] (* Jayanta Basu, May 26 2013 *)

PARI

A205510(n)=norml2(binary(bitxor(prime(n),prime(n+1))))  \\ M. F. Hasler, Jan 29 2012

PARI

a(n,p=prime(n),q=nextprime(p+1))=hammingweight(bitxor(p,q)) \\ Charles R Greathouse IV, Nov 15 2022

Extensions

Corrected a(24) and a(25) by M. F. Hasler, Jan 29 2012

Added "binary" to definition. - N. J. A. Sloane, Jul 09 2021