A205510 - OEIS

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

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OFFSET

1,2

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

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 *)

PROG

(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

CROSSREFS

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