A038371 Smallest prime factor of 10^n+1.

2, 11, 101, 7, 73, 11, 101, 11, 17, 7, 101, 11, 73, 11, 29, 7, 353, 11, 101, 11, 73, 7, 89, 11, 17, 11, 101, 7, 73, 11, 61, 11, 19841, 7, 101, 11, 73, 11, 101, 7, 17, 11, 29, 11, 73, 7, 101, 11, 97, 11, 101, 7, 73, 11, 101, 11, 17, 7, 101, 11, 73, 11, 101, 7, 1265011073

Offset

0,1

Comments

a(n) >= 7 for all n >= 1 since 10^n + 1 is then not divisible by 2, 3 or 5.

Record values are a({0, 1, 2, 16, 32, 64, ...}). - M. F. Hasler, Apr 04 2008

References

Ehrhard Behrends, Five-Minute Mathematics, translated by David Kramer. American Mathematical Society (2008) p. 7

Links

Max Alekseyev, Table of n, a(n) for n = 0..1023 (terms 0..500 from M. F. Hasler)

Makoto Kamada, Factorizations of 100...001.

Formula

a(n) = A020639(A062397(n)).

For odd n, a(n) <= 11 since every (base 10) palindrome of even length is divisible by 11. - M. F. Hasler, Apr 04 2008

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

Example

a(12) = 73 as 10^12+1 = 1000000000001 = 73*137*99990001.

Mathematica

Table[FactorInteger[10^n + 1][[1, 1]], {n, 0, 49}] (* Alonso del Arte, Oct 21 2011 *)

Prog

(PARI) A038371(n)=factor(10^n+1)[1, 1] \\ For large n it may be *much* more efficient to use a(n)=A020639(10^n+1) with less naive code found there. - M. F. Hasler, Apr 04 2008
(Magma) [Min(PrimeFactors(10^n+1)):n in[0..70]]; // Vincenzo Librandi, Nov 08 2018

Crossrefs

Cf. A001221, A003021, A038371, A057934, A062397, A102050, A119704, A185121.

Sequence in context: A056732 A157715 A001271 * A236174 A003021 A097463

Adjacent sequences: A038368 A038369 A038370 * A038372 A038373 A038374

Keyword

nonn

Author

Miklos SZABO (mike(AT)ludens.elte.hu)

Extensions

More terms from Reinhard Zumkeller, Mar 12 2002

Status

approved