A294656 -id:A294656 - cp's OEIS frontend

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

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

A294657 Largest number in the orbit of n under iteration of the map A125256: x -> smallest odd prime divisor of n^2+1.

Original entry on oeis.org

13, 13, 17, 13, 37, 13, 13, 421, 5101, 1861, 13, 13, 197, 113, 257, 17, 18, 16381, 401, 21, 22, 23, 577, 313, 677, 27, 28, 421, 30, 31, 32, 33, 34, 613, 1297, 37, 38, 761, 1601, 421, 42, 43, 44, 1013, 421, 47, 48, 1201, 421, 1301, 52, 53, 2917, 55, 3137, 57, 58, 1515541, 60

Offset: 2

M. F. Hasler, Nov 06 2017

nonn

The orbit (or trajectory) under A125256 appears to end in the cycle 5 -> 13 -> 5 -> etc. for any initial value n.

Ray Chandler, Table of n, a(n) for n = 2..20001

Cf. A125256, A294656 (size of the orbit).

Table[Max[NestWhileList[SelectFirst[FactorInteger[#^2+1][[All,1]], OddQ]&, n,#!=13&]],{n,2,60}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 04 2019 *)

A294657(n,S=[n])={while(#S<#S=setunion(S,[n=A125256(n)]),); vecmax(S)}

A294658 Number of steps required to reach either 5 or 13, starting with n, when iterating the map A125256: x -> smallest odd prime divisor of n^2+1; or a(n) = -1 in case 5 is never reached.

Original entry on oeis.org

1, 1, 2, 0, 2, 1, 1, 4, 3, 5, 1, 0, 2, 2, 2, 1, 1, 4, 3, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 4, 3, 1, 1, 2, 2, 2, 1, 1, 4, 1, 4, 1, 1, 2, 2, 2, 1, 1, 5, 1, 3, 1, 1, 2, 3, 2, 1, 1, 4, 2, 2, 1, 1, 2, 2, 1, 1

Offset: 2

M. F. Hasler, Nov 06 2017

nonn

The orbit or trajectory under A125256 appears to end in the cycle 5 -> 13 -> 5 -> etc. for any initial value n.

Sequence A294656 gives the size of the complete orbit of n under the map A125256, including the two elements 5 and 13 of the terminating cycle. Thus a(n) is 2 less than A294656(n) for all n. This is confirmed by careful examination of special cases - assuming, of course, that all trajectories end in the cycle (5, 13).

			For n = 1 the map A125256 is not defined.
a(2) = 1 because under A125256, 2 -> 2^2+1 = 5 (= its smallest odd prime factor), so 5 is reached after just a(2) = 1 iteration of this map.
a(3) = 1 because A125256(3) = 5, least odd prime factor of 3^2+1 = 10 = 2*5, so here again 5 is reached after just a(2) = 1 iteration of A125256.
a(4) = 2 because A125256(4) = 4^2 + 1 = 17, and A125256(17) = 5 = least odd prime factor of 17^2 + 1 = 289 + 1 = 2*5*29, so 5 is reached after a(4) = 2 iterations of A125256.
a(5) = a(13) = 0 because for these initial values 5 and 13, no iteration is needed until either 5 or 13 is reached.

Ray Chandler, Table of n, a(n) for n = 2..20001

Cf. A125256, A294656, A294657.

A294658(n)=for(k=0,oo,bittest(8224,n)&&return(k);n=A125256(n)) \\ 8224 = 2^5 + 2^13. One could add 2^0 + 2^1 = 3 to avoid an error message for initial values 0 and 1, for which A125256 is not defined.

a(n) = A294656(n) - 2.

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

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A294657 Largest number in the orbit of n under iteration of the map A125256: x -> smallest odd prime divisor of n^2+1.

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A294658 Number of steps required to reach either 5 or 13, starting with n, when iterating the map A125256: x -> smallest odd prime divisor of n^2+1; or a(n) = -1 in case 5 is never reached.

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