A345985 -id:A345985 - cp's OEIS frontend

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

Vladimir Shevelev, Jan 28 2012

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.

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

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

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

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

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

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

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

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

A353737 Length of longest n-digit optimal prime ladder (base 10).

Original entry on oeis.org

2, 4, 7, 9, 11, 13, 15

Offset: 1

Michael S. Branicky, May 09 2022

A prime ladder (in base b) starts with a prime, ends with a prime, and each step produces a new prime by changing exactly one base-b digit.
A shortest such construct between two given primes is optimal.
Analogous to a word ladder (see Wikipedia link).
Here, n-digit primes do not allow leading 0 digits.
If all n-digit primes are disconnected, a(n) = 1; if there are no n-digit primes, a(n) = 0.
a(7) >= 15.
It follows from Bertrand's postulate that there exist n-digit primes for all n >= 1, so a(n) is never 0. - Pontus von Brömssen, May 11 2022

			The 1-digit optimal prime ladder 3 - 5 is tied for the longest amongst 1-digit primes, so a(1) = 2.
The 2-digit optimal prime ladder 97 - 17 - 13 - 53 is tied for the longest amongst 2-digit primes, so a(2) = 4.
The 3-digit optimal prime ladder 389 - 383 - 283 - 281 - 251 - 751 - 761 is tied for the longest amongst 3-digit primes, so a(3) = 7.
The 4-digit optimal prime ladder 4651 - 4951 - 4931 - 4933 - 4733 - 6733 - 6833 - 6883 - 6983 is tied for the longest amongst 4-digit primes, so a(4) = 9.
The 5-digit optimal prime ladder 88259 - 48259 - 45259 - 45959 - 41959 - 41969 - 91969 - 91961 - 99961 - 99761 - 99721 is tied for the longest amongst 5-digit primes, so a(5) = 13.
The 6-digit optimal prime ladder 440497 - 410497 - 410491 - 710491 - 710441 - 710443 - 717443 - 917443 - 917843 - 907843 - 905843 - 905833 - 995833 is tied for the longest amongst 6-digit primes, so a(6) = 13.
The 7-digit optimal prime ladder 3038459 - 3032459 - 3032453 - 3034453 - 3034457 - 3034657 - 3074657 - 3074557 - 4074557 - 4079557 - 4779557 - 4779547 - 7779547 - 7759547 - 7755547 is tied for the longest amongst 7-digit primes, so a(7) = 15. - _Michael S. Branicky_, May 21 2022

Wikipedia, Word Ladder
Zach Wissner-Gross, Can You Build The Longest Ladder?, Riddler Classic, May 06 2022.

Cf. A000040, A050249, A125002, A158576, A253269, A345985.

a(n) is the number of vertices of a longest shortest path in the graph G = (V, E), where V = {n-digit base-10 primes} and E = {(v, w) | H_10(v, w) = 1}, where H_b is the Hamming distance in base b.

a(7) from Michael S. Branicky, May 21 2022

cp's OEIS Frontend

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

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