A055386 -id:A055386 - cp's OEIS frontend

A055385 Smallest prime factor of n^n + 1.

2, 5, 2, 257, 2, 13, 2, 97, 2, 101, 2, 89, 2, 29, 2, 274177, 2, 5, 2, 148721, 2, 5, 2, 17, 2, 53, 2, 449, 2, 17, 2, 641, 2, 13, 2, 17, 2, 5, 2, 17, 2, 5, 2, 41, 2, 29, 2, 769, 2, 41, 2, 89, 2, 13, 2, 17, 2, 5, 2, 17, 2, 5, 2, 59649589127497217, 2, 37, 2, 41, 2, 13, 2, 97, 2, 149

Offset: 1

Walter Nissen, Jun 24 2000

nonn

If we use the commonly accepted convention that 0^0 = 1, then a(0) = 2. - Chai Wah Wu, Jul 22 2019

			4^4 + 1 = 257 prime, so a(4) = 257;
6^6 + 1 = 13 * 37 * 97, so a(6) = 13.

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory. Dover. New York: 1988. Page 82.

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A014566, A055386.

Table[FactorInteger[n^n + 1][[1, 1]], {n, 74}] (* Vincenzo Librandi, Jul 23 2013 *)

a(n) = {if (n % 2, return (2)); return (factor(n^n + 1)[1, 1]);} \\ Michel Marcus, Jul 23 2013

A076670 Prime divisors of (10^9)^(10^9) + 1 = 10^9000000000 + 1.

Original entry on oeis.org

39937, 64513, 921601, 1514497, 9188353, 11059201, 23500801, 25159681, 99328001, 288000001, 302078977, 593920001, 864000001, 14400000001, 16002416641, 27769098241, 35904000001, 61120000001, 61600000001, 90708480001, 164457013249

Offset: 1

Donald S. McDonald, Oct 25 2002

Numbers of the form 10^{10h}+1 can be algebraically factored into (10^{2h}+1)*L*M, L=A-B, M=A+B, h=2k-1, A=10^{4h}+5.10^{3h}+7.10^{2h}+5.10^h+1, B=10^k(10^{3h}+2.10^{2h}+2.10^h+1).
Cyclotomic factorization: 10^(9*10^9) + 1 = Product_{d|9*5^9} Phi_{1024*d}(10).
Every term is congruent to 1, 2049, 3073, or 9217 modulo 10240. - Max Alekseyev, Aug 30 2023
a(32) > 10^16. - Max Alekseyev, Sep 10 2023
Contains 1137797098931682858642433, 3611707318387778163302401. - Max Alekseyev, Sep 10 2023

			a(1) = 39937 because 39937 is the smallest prime divisor of (10^9)^(10^9) + 1.

NZ Science Monthly Bulletin Board, advert., 2000.

Max Alekseyev, Table of n, a(n) for n = 1..31
S. S. Wagstaff, The Cunningham Project.

Cf. A055386 (least prime factor of (2n)^(2n) + 1 ).

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 2; Do[ While[ PowerMod[10, 9000000000, p] + 1 != p, p = NextPrim[p]]; Print[p]; p++, {n, 1, 19}]

forstep(p=1, 10^14, 1024, if(!ispseudoprime(p), next); if(Mod(10,p)^9000000000==-1, print(p)); )

Thanks for help from Kurt Foster and Bob Backstrom (Australia) - Donald S. McDonald
Edited and extended by Robert G. Wilson v, Nov 13 2002
Definition corrected by Sean A. Irvine, Feb 16 2010
Definition corrected by Max Alekseyev, Apr 28 2010
a(20)-a(31) from Max Alekseyev, Apr 28 2013, Jul 02 2013, Sep 10 2023

cp's OEIS Frontend

A055385 Smallest prime factor of n^n + 1.

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