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Binary representations of n and n^2 have no common 3-digit substring.
1

%I #22 Sep 02 2019 04:55:47

%S 0,1,2,3,5,6,7,11,23,46,85,341,1471,2942,5461,21845,349525,1398101,

%T 6025215,12050430,22369621,89478485,1431655765,5726623061,91625968981,

%U 366503875925,5864062014805,23456248059221,101086333501439,202172667002878,375299968947541,1501199875790165

%N Binary representations of n and n^2 have no common 3-digit substring.

%C n such that A177062(n) <= 2.

%C Contains (4^m-1)/3 for m not divisible by 3.

%C Contains 2^(3*2^(k-1)-1) - 1 - Sum_{j=1..k-1} 2^(3*(2^(k-1)-2^(j-1))) and 2^(3*2^(k-1)) - 2 - Sum_{j=1..k-1} 2^(3*(2^(k-1)-2^(j-1))+1) for all k>=1.

%C Are there any other terms besides these and 0, 2, 7, 11?

%e The binary representations of 11 and 11^2 are 1011 and 1111001. The 3-digit substrings 101 and 011 of 1011 are not substrings of 1111001, so 11 is in the sequence.

%p f:= n -> length(StringTools:-LongestCommonSubString(convert(convert(n,binary),string),convert(convert(n^2,binary),string))):

%p select(f <= 2, [$0..1000]);

%Y Cf. A177062.

%K nonn,base

%O 1,3

%A _Robert Israel_, Sep 13 2016

%E a(27)-a(32) from _Giovanni Resta_, Aug 31 2019