A394434 a(n) is the smallest prime factor of Sum_{k=0..n} 10^2k

101, 3, 73, 41, 3, 239, 17, 3, 41, 11, 3, 53, 29, 3, 17, 103, 3, 909090909090909091, 41, 3, 11, 47, 3, 41, 53, 3, 29, 59, 3, 2791, 17, 3, 101, 41, 3, 7253, 101, 3, 17, 83, 3, 173, 11, 3, 47, 6299, 3, 197, 41, 3, 53, 107, 3, 11, 17, 3, 59, 1889, 3, 733, 101, 3

Offset

1,1

Links

Michael S. Branicky, Table of n, a(n) for n = 1..344 (using factordb.com)

factordb.com, Status of (10^(2*n+2)-1)/99.

Formula

a(n) = 3 if n mod 3 == 2.

a(n) = A020639((10^(2*n+2)-1)//99).

a(n) = A020639(A094028(n)). - Michael S. Branicky, Apr 18 2026

Example

a(1) = 101 since 101 is prime.
a(2) = 3 since 10101 = 3*7*13*37.
a(3) = 73 since 1010101 = 73*101*137.
a(4) = 41 since 101010101 = 41*271*9091.

Prog

(Python)
from sympy import factorint
def a(n): return min(factorint((10**(2*n+2)-1)//99))
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Apr 18 2026
(Julia)
function a(n)
    num = (big(10)^(2n + 2) - 1) ÷ 99
    for i in 3:2:num
        if num % i == 0
            return i
        end
    end
end # Hoang Nguyen, Apr 21 2026

Crossrefs

Cf. A020639, A094028, A016789.

Sequence in context: A036193 A126834 A365928 * A318698 A180236 A279149

Adjacent sequences: A394431 A394432 A394433 * A394435 A394436 A394437

Keyword

nonn

Author

Hoang Nguyen, Apr 16 2026

Extensions

a(9) onward from Michael S. Branicky, Apr 18 2026

Status

approved