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%I #102 Nov 15 2023 09:31:21
%S 2,7,3,11,41,3,61,7,3,11,113,3,53,7,3,11,29,3,17,7,3,11,
%T 11111111111111111111111,3,41,7,3,11,53,3,661,7,3,11,17,3,2028119,7,3,
%U 11,83,3,173,7,3,11,40697,3,239,7,3,11,107,3,41,7,3,11,2836549,3,733,7,3,11
%N Smallest prime factor of f(n) = 10^(2*n+1) + (10^n-1)/9.
%C f(n) = 100..00011..11 is the least positive integer whose decimal digits are n+1 1's and n+1 0's.
%H Jean-Marc Rebert, <a href="/A365966/b365966.txt">Table of n, a(n) for n = 0..309</a>
%F a(n) = 3 iff n = 3k + 2, since f(n) is odd and has n+1 1 digits so that "casting out 9's" shows f(n) == n+1 (mod 3).
%F a(n) = 7 iff n = 6k + 1.
%F a(n) = 11 iff n = 6k + 3.
%e a(1) = 7, because the smallest prime factor of f(1) = 1001 = 7 * 11 * 13 is 7.
%e a(2) = 3, because the smallest prime factor of f(2) = 100011 = 3 * 17 * 37 * 43 is 3.
%t a[n_]:=Min[First/@FactorInteger[10^(2*n+1)+(10^n-1)/9]]; Array[a,64,0] (* _Stefano Spezia_, Sep 24 2023 *)
%o (PARI) a365966(n, limtd=10^9) = {my (x=10^(2*n+1)+(10^n-1)/9); forprime (p=2, limtd, if(x%p==0, return(p))); factor(x)[1,1]}; \\ _Hugo Pfoertner_, Nov 14 2023
%Y Cf. A020639, A365928, A365610, A067063.
%K nonn
%O 0,1
%A _Jean-Marc Rebert_, Sep 23 2023