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

0,1

Comments

f(n) = 100..00011..11 is the least positive integer whose decimal digits are n+1 1's and n+1 0's.

Links

Jean-Marc Rebert, Table of n, a(n) for n = 0..309

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.

Example

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.

Mathematica

a[n_]:=Min[First/@FactorInteger[10^(2*n+1)+(10^n-1)/9]]; Array[a, 64, 0] (* Stefano Spezia, Sep 24 2023 *)

Prog

(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

Crossrefs

Cf. A020639, A365928, A365610, A067063.

Sequence in context: A091578 A332363 A353408 * A258249 A256448 A056756

Adjacent sequences: A365963 A365964 A365965 * A365967 A365968 A365969

Keyword

nonn

Author

Jean-Marc Rebert, Sep 23 2023

Status

approved