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%I #49 Jun 30 2024 09:10:30
%S 2,11,101,7,73,11,101,11,17,7,101,11,73,11,29,7,353,11,101,11,73,7,89,
%T 11,17,11,101,7,73,11,61,11,19841,7,101,11,73,11,101,7,17,11,29,11,73,
%U 7,101,11,97,11,101,7,73,11,101,11,17,7,101,11,73,11,101,7,1265011073
%N Smallest prime factor of 10^n + 1.
%C a(n) >= 7 for all n >= 1 since 10^n + 1 is then not divisible by 2, 3 or 5.
%C Record values are a({0, 1, 2, 16, 32, 64, ...}). - _M. F. Hasler_, Apr 04 2008
%C The record values (2, 11, 101, 353, 19841, 1265011073, ...) are also found in A185121 and A102050 (smallest prime factor of 10^2^n+1). - _M. F. Hasler_, Jun 28 2024
%D Ehrhard Behrends, Five-Minute Mathematics, translated by David Kramer. American Mathematical Society (2008) p. 7
%H Max Alekseyev, <a href="/A038371/b038371.txt">Table of n, a(n) for n = 0..1023</a> (terms 0..500 from M. F. Hasler)
%H Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit/10001.htm">Factorizations of 100...001</a>.
%F a(n) = A020639(A062397(n)).
%F For odd n, a(n) <= 11 since every (base 10) palindrome of even length is divisible by 11. - _M. F. Hasler_, Apr 04 2008 [See below for more precise formula.]
%F More generally, for k >= 0 and n == 2^k (mod 2^(k+1)), a(n) <= A185121(k) = (11, 101, 73, 17, 353, ...). This follows from x^{2q+1} + 1 = (x+1) Sum_{m=0..2q} (-x)^m, with x=10^2^k. - _M. F. Hasler_, Jul 30 2019
%F From _M. F. Hasler_, Jun 28 2024: (Start)
%F a(2k+1) = 7 iff k == 1 (mod 3), else 11. [Making the 2008 formula more precise.]
%F a(4k+2) = 29 iff k == 3 (mod 7), else = 61 if k == 7 (mod 15), else = 89 if k == 5 (mod 11), else 101.
%F a(8k+4) = 73 for all k >= 0.
%F a(16k+8) = 17 for all k >= 0.
%F a(32k+16) = 97 iff k==1 (mod 3), else 353.
%F a(64k+32) = 193 iff k==1 (mod 3), else 1217 if k==9 (mod 19), else 2753 if k==21 (mod 43), else 3137 if k==24 (mod 49), else 3329 if k==6 (mod 13), else 4481 if k==17 (mod 35), else 4673 if k==36 (mod 73), else 5953 if k==15 (mod 31), else 6529 if k==8 (mod 17), else 13633 if k==35 (mod 71), else 15937 if k==41 (mod 83), else 19841. (End)
%e a(12) = 73 as 10^12+1 = 1000000000001 = 73*137*99990001.
%t Table[FactorInteger[10^n + 1][[1, 1]], {n, 0, 49}] (* _Alonso del Arte_, Oct 21 2011 *)
%o (PARI) A038371(n)=A020639(10^n+1) \\ Much more efficient than the naive {factor(10^n+1)[1,1]}. - _M. F. Hasler_, Apr 04 2008, edited Jun 29 2024
%o (Magma) [Min(PrimeFactors(10^n+1)):n in[0..70]]; // _Vincenzo Librandi_, Nov 08 2018
%Y Cf. A020639 (least prime factor), A062397 (10^n + 1), A003021 (largest prime factor of 10^n + 1), A057934 (number of prime factors of 10^n + 1, with multiplicity), A119704 (as before, without multiplicity), A185121 (smallest prime factor of 10^2^n+1), A102050 (as before, but 1 if 10^2^n+1 is prime).
%K nonn
%O 0,1
%A Miklos SZABO (mike(AT)ludens.elte.hu)
%E More terms from _Reinhard Zumkeller_, Mar 12 2002
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