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A086982 Numbers n such that 10^n+1 is not squarefree.

%I A086982 #21 Nov 01 2024 09:35:24
%S A086982 11,21,33,39,55,63,77,99,105,117,121,136,143,147,165,171,187,189,195,
%T A086982 202,209,231,243,253,273,275,292,297,315,319,341,351,357,363,385,399,
%U A086982 406,407,408,429,441,451,473,483,495,507,513,517,525,539,548,561,567
%N A086982 Numbers n such that 10^n+1 is not squarefree.
%C A086982 This sequence is the union of the collection of sequences formed from the nonzero terms of A086981 * A005408, the odd numbers. First occurrence of consecutive integers in sequence is 406,407,408.
%C A086982 From _Robert Israel_, Feb 13 2017: (Start)
%C A086982 Numbers n such that gcd(n, 10^n + 1) > 1 or n = k*m where k is odd and 2*m is the order of 10 modulo a member of A045616. [Corrected by _Jianing Song_, Nov 01 2024]
%C A086982 If n is in the sequence, then so is k*n for any odd k. (End)
%C A086982 Numbers of the form k*ord(10,p^2)/2, where k is an odd number and p is a prime such that ord(10,p) is even. Here ord(a,m) is the multiplicative order of a modulo m. Note that if p is not in A045616, then ord(10,p^2) = p*ord(10,p). - _Jianing Song_, Nov 01 2024
%H A086982 Robert Israel, <a href="/A086982/b086982.txt">Table of n, a(n) for n = 1..10000</a>
%p A086982 filter:= n -> (n mod 243 = 0 and (n/243)::odd) or igcd(n,(10 &^n +1 mod n)) > 1: # Note that this works if n < 28299156
%p A086982 select(filter, [$1..1000]); # _Robert Israel_, Feb 13 2017
%o A086982 (PARI) ord = [1, 486, 56598312]; \\ order of 10 modulo A045616
%o A086982 isA086982(n) = if(gcd(n, 10^n+1) > 1, return(1)); for(i=1, 3, if((ord[i] % 2 == 0) && (n % (ord[i]/2) == 0) && (n/(ord[i]/2) % 2 == 1), return(1))); return(0) \\ _Jianing Song_, Nov 01 2024, after _Robert Israel_'s comment; considering only the three currently-known terms of A045616
%Y A086982 Cf. A005408, A013929, A086981, A045616, A049096, A282344, A282345.
%K A086982 nonn
%O A086982 1,1
%A A086982 _Ray Chandler_, Jul 27 2003