A365966 Smallest prime factor of f(n) = 10^(2*n+1) + (10^n-1)/9.
2, 7, 3, 11, 41, 3, 61, 7, 3, 11, 113, 3, 53, 7, 3, 11, 29, 3, 17, 7, 3, 11, 11111111111111111111111, 3, 41, 7, 3, 11, 53, 3, 661, 7, 3, 11, 17, 3, 2028119, 7, 3, 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
Offset: 0
Keywords
Examples
a(1) = 7, because the smallest prime factor of f(1) = 1001 = 7 * 11 * 13 is 7. a(2) = 3, because the smallest prime factor of f(2) = 100011 = 3 * 17 * 37 * 43 is 3.
Links
- Jean-Marc Rebert, Table of n, a(n) for n = 0..309
Programs
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Mathematica
a[n_]:=Min[First/@FactorInteger[10^(2*n+1)+(10^n-1)/9]]; Array[a,64,0] (* Stefano Spezia, Sep 24 2023 *)
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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
Formula
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).
a(n) = 7 iff n = 6k + 1.
a(n) = 11 iff n = 6k + 3.
Comments