A253269 -id:A253269 - cp's OEIS frontend

A158576 a(n) = number of components of the graph P(n,10) (defined in Comments).

1, 1, 1, 1, 1, 7, 38, 365, 3355, 33586

Offset: 1

W. Edwin Clark, Mar 21 2009

Let H(n,b) be the Hamming graph whose vertices are the sequences of length n over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(n,b) be the subgraph of H(n,b) induced by the set of vertices which are base b representations of primes with n digits (not allowing leading 0 digits).
For 6 and 7 digit primes there is a single large component and the remaining components have size 1. For 8 digit primes there is a single large component, the size 2 component {89391959, 89591959} and the remaining components have size 1. - W. Edwin Clark, Mar 31 2009
The elements of size 2 components in these graphs are sequence A253269. - Michael Kleber, May 04 2015

			The 6-digit primes 294001, 505447, 584141, 604171, 929573, 971767 (cf. A050249) have the property that changing any single digit always gives a composite number, so these are isolated nodes in the graph P(6,10) (which also has one large connected component).

Cf. A050249, A145667, A145668, A145669, A145670, A145671, A145672, A145673, A145674, A158577, A158578, A158579, A253269.

a(8) from W. Edwin Clark, Mar 31 2009

a(9)-a(10) from Max Alekseyev, Dec 23 2024

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

A158576 a(n) = number of components of the graph P(n,10) (defined in Comments).

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