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A078813 Smallest prime factor of googol - n that exceeds 13, or 1 if googol - n is 13-smooth.

%I A078813 #49 Jul 28 2025 19:44:46
%S A078813 1,41,220217,596275259857,17,31,7583,167988019,1898431,19,37,8747,433,
%T A078813 23,4647535350279428239,1637,29,1997,569,383,71,17,179,
%U A078813 683592593118601,601,1259,109,47,19,83,367,43,151,8633431,103,20859069935591,23
%N A078813 Smallest prime factor of googol - n that exceeds 13, or 1 if googol - n is 13-smooth.
%H A078813 Sean A. Irvine, <a href="/A078813/b078813.txt">Table of n, a(n) for n = 0..1000</a>
%H A078813 Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/GOOGOL.HTM">Factors of 1000 numbers starting from googolplex</a>
%H A078813 Robert Harley, <a href="https://pauillac.inria.fr/~harley/bits.html">Factors of Googolplex+1</a>
%H A078813 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Googol.html">Googol</a>
%F A078813 For n >= 1, a(n) = A020639(A007947(10^100 - n)/gcd(10^100 - n, A034386(13))), where A020639(m) = lpf(m), smallest prime factor of m. - _Peter Munn_, Feb 20 2025
%F A078813 a(-n) = A076848(n). - _Zhuorui He_, Jul 15 2025
%e A078813 From _Zhuorui He_, Jul 15 2025: (Start)
%e A078813 Googol = 10^100 = 2^100 * 5^100 is 13-smooth so a(0)=1.
%e A078813 10^100 - 1 = 3^2 * 11 * 41 * 101 * 251 * 271 * ... so a(1)=41. (End)
%o A078813 (PARI) /* using M. F. Hasler's definition for A020639 */
%o A078813 A078813(n)={n=10^100-n; my(p=[2,3,5,7,11,13]); for(i=1, 6, n=n/(p[i]^valuation(n,p[i]))); A020639(n)} /* _Zhuorui He_ , Jul 17 2025 */
%Y A078813 Cf. A108251 (n such that googol - n is prime), A080197 (relates to positions of 1's).
%Y A078813 Equivalent sequences: A076848 (googol + n), A078814 (googolplex - n).
%Y A078813 See the formula section for the relationships with A007947, A020639, A034386.
%K A078813 nonn
%O A078813 0,2
%A A078813 _Robert G. Wilson v_, Dec 06 2002
%E A078813 Name edited by _Peter Munn_, Feb 20 2025
%E A078813 a(0) prepended by _Zhuorui He_, Jul 15 2025