A293958 -id:A293958 - cp's OEIS frontend

5, 5, 17, 13, 37, 5, 5, 41, 101, 61, 5, 5, 197, 113, 257, 5, 5, 181, 401, 13, 5, 5, 577, 313, 677, 5, 5, 421, 17, 13, 5, 5, 13, 613, 1297, 5, 5, 761, 1601, 29, 5, 5, 13, 1013, 29, 5, 5, 1201, 41, 1301, 5, 5, 2917, 17, 3137, 5, 5, 1741, 13, 1861, 5, 5, 17, 2113, 4357, 5, 5

Offset: 2

Nick Hobson, Nov 26 2006

Any odd prime divisor of n^2+1 is congruent to 1 modulo 4.
n^2+1 is never a power of 2 for n > 1; hence a prime divisor congruent to 1 modulo 4 always exists.
a(n) = 5 if and only if n is congruent to 2 or -2 modulo 5.
If the map "x -> smallest odd prime divisor of n^2+1" is iterated, does it always terminate in the 2-cycle (5 <-> 13)? - Zoran Sunic, Oct 25 2017

			The prime divisors of 8^2 + 1 = 65 are 5 and 13, so a(7) = 5.

D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 191.

Ray Chandler, Table of n, a(n) for n = 2..20001 (first 999 terms from Nick Hobson)

Cf. A002522, A002496, A014442, A057207, A031439.
For bisections see A256970, A293958.
See also A125257, A125258, A294656, A294657, A294658.

with(numtheory, factorset);
A125256 := proc(n) local t1,t2;
if n <= 1 then return(-1); fi;
if (n mod 5) = 2 or (n mod 5) = 3 then return(5); fi;
t1 := numtheory[factorset](n^2+1);
t2:=sort(convert(t1,list));
if (n mod 2) = 1 then return(t2[2]); fi;
t2[1];
end;
[seq(A125256(n),n=1..40)]; # N. J. A. Sloane, Nov 04 2017

Table[Select[First/@FactorInteger[n^2+1],OddQ][[1]],{n,2,68}] (* James C. McMahon, Dec 16 2024 *)

vector(68, n, if(n<2, "-", factor(n^2+1)[1+(n%2),1]))

A125256(n)=factor(n^2+1)[1+bittest(n,0),1] \\ M. F. Hasler, Nov 06 2017

cp's OEIS Frontend

A125256 Smallest odd prime divisor of n^2 + 1.

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