A125256 -id:A125256 - cp's OEIS frontend

A294656 Size of the orbit of n under iteration of the map A125256: x -> smallest odd prime divisor of n^2+1.

3, 3, 4, 2, 4, 3, 3, 6, 5, 7, 3, 2, 4, 4, 4, 3, 3, 6, 5, 3, 3, 3, 4, 4, 4, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 3, 3, 5, 5, 5, 3, 3, 3, 4, 5, 3, 3, 4, 6, 5, 3, 3, 4, 4, 4, 3, 3, 6, 3, 6, 3, 3, 4, 4, 4, 3, 3, 7, 3, 5, 3, 3, 4, 5, 4, 3, 3, 6, 4, 4, 3, 3, 4, 4, 3, 3, 3, 4, 8, 6, 3

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 A294658 gives the number of steps to reach either 5 or 13, i.e. an element of this terminating cycle. Therefore a(n) (which counts these two elements as well as the initial value) is 2 more than A294658(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) = 3 = # { 2, 5, 13 }, because under A125256, 2 -> 2^2+1 = 5 (= its smallest odd prime factor), 5 -> least odd prime factor(5^2+1 = 26) = 13, 13 -> least odd prime factor(13^2 + 1 = 170 = 2*5*17) = 5, etc.
a(3) = 3 = # { 3, 5, 13 }, because under A125256, 3 -> smallest odd prime factor(3^2+1 = 10) = 5, 5 -> 13, 13 -> 5 etc.
a(4) = 4 = # { 4, 17, 5, 13 }, because under A125256, 4 -> 4^2+1 = 17 (= its smallest odd prime factor), 17 -> smallest odd prime factor(17^2+1 = 290 = 2*5*29) = 5, 5 -> 13, 13 -> 5 etc.

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

Cf. A125256, A294657: largest number in the orbit, A294658: number of steps to reach the cycle (5, 13).

A294656(n,f=A125256,S=[n])={while(#S<#S=setunion(S,[n=f(n)]),); #S} \\ Does not assume the terminating cycle is (5, 13): also works correctly in case there are other terminating cycles.

a(n) = A294658(n) + 2.

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.

A031439 a(0) = 1, a(n) is the greatest prime factor of a(n-1)^2+1 for n > 0.

Original entry on oeis.org

1, 2, 5, 13, 17, 29, 421, 401, 53, 281, 3037, 70949, 1713329, 1467748131121, 37142837524296348426149, 101591133424866642486477019709, 1650979973845742266714536305651329, 78343914631785958284737, 4029445531112797145738746391569, 350080544438648120162733678636001, 26208090024628793745288451837610346882122253572537, 4717815978577117335515270825550279551117660519482308365269206484133871485221

Offset: 0

Yasutoshi Kohmoto

Does this sequence grow indefinitely, or does it cycle? - Franklin T. Adams-Watters, Oct 02 2006
All a(n) except a(0) = 1 belong to A014442(n) = {2, 5, 5, 17, 13, 37, 5, 13, 41, 101, ...} Largest prime factor of n^2 + 1. All a(n) except a(0) = 1 belong to A002313(n) = {2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, ...} Primes congruent to 1 or 2 modulo 4; or, primes of form x^2+y^2; or, -1 is a square mod p. All a(n) except a(0) = 1 and a(1) = 2 are the Pythagorean primes A002144(n) = {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, ...} Primes of form 4n+1. - Alexander Adamchuk, Nov 05 2006
Essentially the same as A072268; A072268(n) = A031439(n-1)^2 + 1. - Charles R Greathouse IV, May 08 2009

			a(16)=A006530(a(15)^2+1)=
A006530(101591133424866642486477019709^2+1)=
A006530(10320758390549056348725939119133160378521185060950774444682)=
A006530(2*29*23201*4645528280970018601*1650979973845742266714536305651329)=
1650979973845742266714536305651329, factorization of A006530(a(15)^2+1) by Dario A. Alpern's program (see link).

Dennis Langdeau, Table of n, a(n) for n = 0..24
Dario A. Alpern, Factorization: Elliptic Curve Method
Jason Papadopoulos, Integer Factorization Source Code.

Cf. A056650, A003095.
Cf. A002144 - Pythagorean primes: primes of form 4n+1; A002313 - Primes congruent to 1 or 2 modulo 4; A014442 - Largest prime factor of n^2 + 1.
Cf. also A072268, A056650, A003095, A125256.

gpf[n_] := FactorInteger[n][[-1, 1]]; a[0] = 1; a[n_] := a[n] = gpf[a[n - 1]^2 + 1]; Table[an = a[n]; Print[an]; an, {n, 0, 21}] (* Jean-François Alcover, Nov 04 2011 *)
NestList[FactorInteger[#^2+1][[-1,1]]&,1,21] (* Harvey P. Dale, Jul 04 2013 *)

gpf(n)=local(pf);pf=factor(n);pf[matsize(pf)[1],1] vector(20,i,r=if(i==1,1,gpf(r^2+1)))

One more term from Vladeta Jovovic, Nov 26 2001
a(16) from Reinhard Zumkeller, Aug 07 2004
a(17)-a(21) from Richard FitzHugh (fitzhughrichard(AT)hotmail.com), Aug 12 2004

A256970 Smallest prime divisor of 4*n^2+1.

Original entry on oeis.org

5, 17, 37, 5, 101, 5, 197, 257, 5, 401, 5, 577, 677, 5, 17, 5, 13, 1297, 5, 1601, 5, 13, 29, 5, 41, 5, 2917, 3137, 5, 13, 5, 17, 4357, 5, 13, 5, 5477, 53, 5, 37, 5, 7057, 13, 5, 8101, 5, 8837, 13, 5, 73, 5, 29, 17, 5, 12101, 5, 41, 13457, 5

Offset: 1

N. J. A. Sloane, Apr 19 2015

nonn

a(n) = A020639(A053755(n)).

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

Richard Friedberg, An Adventurer's Guide to Number Theory, McGraw-Hill, NY, 1968.
Popular Computing (Calabasas, CA), Friedberg's Sequence, Vol. 5 (No. 46, Jan 1977), page PC46-2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A053755, A256971, A020639.

A bisection of A125256.

a256970 = a020639 . a053755  -- Reinhard Zumkeller, Apr 20 2015

Table[FactorInteger[4*n^2+1][[1,1]],{n,59}] (* Ivan N. Ianakiev, Apr 20 2015 *)

a(n) = factor(4*n^2+1)[1,1]; \\ Michel Marcus, Apr 20 2015

A293958 Smallest odd prime divisor of (2n+1)^2 + 1.

Original entry on oeis.org

5, 13, 5, 41, 61, 5, 113, 5, 181, 13, 5, 313, 5, 421, 13, 5, 613, 5, 761, 29, 5, 1013, 5, 1201, 1301, 5, 17, 5, 1741, 1861, 5, 2113, 5, 2381, 2521, 5, 29, 5, 3121, 17, 5, 3613, 5, 17, 41, 5, 4513, 5, 13, 5101, 5, 37, 5, 13, 61, 5, 17, 5, 73, 7321, 5, 13, 5, 53, 8581, 5, 13, 5, 9661, 9941, 5

Offset: 1

N. J. A. Sloane, Nov 04 2017, following a suggestion from Zoran Sunic

nonn

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

A027862 is a subsequence. - David A. Corneth, Nov 04 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

A bisection of A125256. Cf. A027862, A069894, A078701, A256970.

sod[n_]:=With[{fi=FactorInteger[n]},If[fi[[1,1]]==2,fi[[2,1]],fi[1,1]]]; sod/@(Range[3,151,2]^2+1) (* Harvey P. Dale, Dec 23 2023 *)

a(n) = factor((2*n+1)^2 + 1)[2,1]; \\ Michel Marcus, Nov 04 2017

a(n) = A078701(A069894(n)). - Michel Marcus, Nov 04 2017

cp's OEIS Frontend

A294656 Size of the orbit of n under iteration of the map A125256: x -> smallest odd prime divisor of n^2+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A031439 a(0) = 1, a(n) is the greatest prime factor of a(n-1)^2+1 for n > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A256970 Smallest prime divisor of 4*n^2+1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A293958 Smallest odd prime divisor of (2n+1)^2 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula