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%I #27 Dec 23 2024 14:11:04
%S 1,1,1,1,1,7,38,365,3355,33586
%N a(n) = number of components of the graph P(n,10) (defined in Comments).
%C 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).
%C 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
%C The elements of size 2 components in these graphs are sequence A253269. - _Michael Kleber_, May 04 2015
%e 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).
%Y Cf. A050249, A145667, A145668, A145669, A145670, A145671, A145672, A145673, A145674, A158577, A158578, A158579, A253269.
%K base,hard,more,nonn
%O 1,6
%A _W. Edwin Clark_, Mar 21 2009
%E a(8) from _W. Edwin Clark_, Mar 31 2009
%E a(9)-a(10) from _Max Alekseyev_, Dec 23 2024