A014442 -id:A014442 - cp's OEIS frontend

A240553 Greatest prime factor of n^10+1.

2, 41, 1181, 61681, 9161, 6781, 4021, 1321, 42521761, 27961, 212601841, 85403261, 641, 1383881, 131381, 4278255361, 63541, 145501, 16936647121, 222361, 920421641, 150901, 272341, 1801385941, 632133361, 208518605101, 47763361, 84961, 470925821, 12109381

Offset: 1

T. D. Noe, Apr 07 2014

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A014442, A081256, A096172, A240548-A240552.

Table[FactorInteger[n^10 + 1][[-1, 1]], {n, 100}]

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

Original entry on oeis.org

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

A223702 Irregular triangle of numbers k such that A002313(n), the n-th prime not congruent to 3 mod 4 is the largest prime factor of k^2 + 1.

Original entry on oeis.org

1, 2, 3, 7, 5, 8, 18, 57, 239, 4, 13, 21, 38, 47, 268, 12, 17, 41, 70, 99, 157, 307, 6, 31, 43, 68, 117, 191, 302, 327, 882, 18543, 9, 32, 73, 132, 278, 378, 829, 993, 2943, 23, 30, 83, 182, 242, 401, 447, 606, 931, 1143, 1772, 6118, 34208, 44179, 85353, 485298

Offset: 1

T. D. Noe, Apr 03 2013

Note that primes of the form 4x+3 are not divisors.

			Irregular triangle:
   p | {k}
-----+---------------------------------
   2 | {1},
   5 | {2, 3, 7},
  13 | {5, 8, 18, 57, 239},
  17 | {4, 13, 21, 38, 47, 268},
  29 | {12, 17, 41, 70, 99, 157, 307},
  37 | {6, 31, 43, 68, 117, 191, 302, 327, 882, 18543},
  41 | {9, 32, 73, 132, 278, 378, 829, 993, 2943}
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..811 (first 22 rows for primes up to 197)
Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355
Filip Najman, List of Publications Page (Adjacent to entry number 7 are links with a data file for the first 22 rows (=811 terms) of this sequence). [As of Dec 2024, the link has the incorrect URL. Should be https://web.math.pmf.unizg.hr/~fnajman/rezplus1.html]

Cf. A002313, A014442, A177979 (first terms), A185389 (last terms), A223705, A285283, A379346 (row lengths), A379347 (row sums).
Cf. A285282, A285523.
Cf. A223701, A223703, A223704 (related tables).

t = Table[FactorInteger[n^2 + 1][[-1,1]], {n, 10^5}]; Table[Flatten[Position[t, Prime[n]]], {n, 13}]

Definition amended by Andrew Howroyd, Dec 22 2024

A285283 Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 4, 9, 15, 22, 32, 41, 57, 74, 94, 120, 156, 192, 232, 278, 325, 381, 448, 521, 607, 704, 811

Offset: 1

Tomohiro Yamada, Apr 16 2017

In other words, x^2 + 1 is A002313(n)-smooth.
Størmer shows that the number of such integers is finite for any n.
a(n) <= 3^n - 2^n follows from Størmer's argument.
a(n) <= (2^n-1)*(A002313(n)+1)/2 is implicit in Lehmer 1964.
Luca 2004 determines all integers x such that x^2 + 1 is 100-smooth, which is pushed to 200 by Najman 2010.

D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57--69.
Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19--24.
Filip Najman, Smooth values of some quadratic polynomials, Glas. Mat. 45 (2010), 347--355. Tables are available in the author's Home Page (gives all 811 numbers x such that x^2+1 has no prime factor greater than 197).
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13 (1967-1968), 177--236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2), 48 pp.

Equivalents for x(x+1): A145604.

Cf. A002313, A014442, A185389, A223702, A285282, A379346 (first differences).

a(13)-a(22) added by Andrew Howroyd, Dec 22 2024

A164314 Largest prime factor of n^2 - 2.

Original entry on oeis.org

2, 7, 7, 23, 17, 47, 31, 79, 7, 17, 71, 167, 97, 223, 127, 41, 23, 359, 199, 439, 241, 31, 41, 89, 337, 727, 23, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 31, 47, 73, 881, 1847, 967, 17, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 47, 3023, 1567, 191

Offset: 2

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A002583, A014442, A069902.

a:= n-> max(numtheory[factorset](n^2-2)):
seq(a(n), n=2..60);  # Alois P. Heinz, Jul 22 2017

Table[FactorInteger[n^2 - 2][[-1, 1]], {n, 2, 57}] (* Michael De Vlieger, Jul 22 2017 *)

a(n) = vecmax(factor(n^2-2)[,1]); \\ Michel Marcus, Jul 22 2017

a(n) = A006530(A008865(n)).

Offset corrected by R. J. Mathar, Aug 21 2009

A240549 Greatest prime factor of n^6+1.

Original entry on oeis.org

2, 13, 73, 241, 601, 97, 181, 109, 6481, 9901, 1117, 20593, 28393, 1033, 3877, 673, 83233, 457, 769, 12277, 3181, 1489, 7549, 577, 390001, 2521, 530713, 47221, 421, 809101, 922561, 1321, 91141, 1249, 5413, 1678321, 144061, 2083693, 2311921, 41941, 1993, 4621

Offset: 1

T. D. Noe, Apr 07 2014

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A014442, A081256, A096172, A240548-A240553.

Table[FactorInteger[n^6 + 1][[-1, 1]], {n, 100}]

A285282 Numbers n such that n^2 + 1 is 13-smooth.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 18, 57, 239

Offset: 1

Tomohiro Yamada, Apr 16 2017

Equivalently: Numbers n such that all prime factors of n^2 + 1 are <= 13.
Since an odd prime factor of n^2 + 1 must be of the form 4m + 1, n^2 + 1 must be of the form 2*5^a*13^b.
This sequence is complete by a theorem of Størmer.
The largest instance 239^2 + 1 = 2*13^4 also gives the only nontrivial solution for x^2 + 1 = 2y^4 (Ljunggren).

			For n = 8, a(8)^2 + 1 = 57^2 + 1 = 3250 = 2*5^3*13.

W. Ljunggren, Zur Theorie der Gleichung x^2 + 1 = 2y^4, Avh. Norsk Vid. Akad. Oslo. 1(5) (1942), 1--27.

A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13 (1967-1968), 177--236.
Ray Steiner, Simplifying the Solution of Ljunggren's Equation X^2 + 1 = 2Y^4, J. Number Theory 37 (1991), 123--132, more accesible proof of Ljunggren's result.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I Nr. 2 (1897), 48 pp.

Cf. A014442, A252493 (n(n+1) instead of n^2 + 1).

Select[Range[1000], FactorInteger[#^2 + 1][[-1, 1]] <= 13&] (* Jean-François Alcover, May 17 2017 *)

for(n=1, 9e6, if(vecmax(factor(n^2+1)[, 1])<=13, print1(n", ")))

from sympy import primefactors
def ok(n): return max(primefactors(n**2 + 1))<=13 # Indranil Ghosh, Apr 16 2017

A240550 Greatest prime factor of n^7+1.

Original entry on oeis.org

2, 43, 547, 113, 449, 197, 911, 5419, 16493, 909091, 1623931, 13063, 22079, 7027567, 10678711, 15790321, 22796593, 32222107, 226871, 10529, 81867661, 86969, 2969, 183458857, 234750601, 59011, 2269, 35771, 574995877, 1118041, 71821, 86171, 219409, 104119, 11831

Offset: 1

T. D. Noe, Apr 07 2014

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A014442, A081256, A096172, A240548-A240553.

Table[FactorInteger[n^7 + 1][[-1, 1]], {n, 100}]

A240551 Greatest prime factor of n^8+1.

Original entry on oeis.org

2, 257, 193, 65537, 11489, 98801, 169553, 673, 21523361, 5882353, 6304673, 260753, 407865361, 16097, 179953, 6700417, 184417, 113607841, 563377, 1505882353, 300673, 3227992561, 623009, 2311681, 29423041, 57734881, 769, 22223646961, 561377, 4855073

Offset: 1

T. D. Noe, Apr 07 2014

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A014442, A081256, A096172, A240548-A240553.

Table[FactorInteger[n^8 + 1][[-1, 1]], {n, 100}]

A240552 Greatest prime factor of n^9+1.

Original entry on oeis.org

2, 19, 37, 109, 5167, 46441, 117307, 87211, 530713, 52579, 590077, 1801, 937, 132049, 811, 38737, 5653, 465841, 236377, 69481, 613, 5966803, 1117, 7561, 6597973, 102966067, 19927, 102547, 10435069, 120871, 1538083, 18837001, 221401, 745903, 612740917, 55117

Offset: 1

T. D. Noe, Apr 07 2014

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A014442, A081256, A096172, A240548, A240549, A240550, A240551, A240553.

Table[FactorInteger[n^9 + 1][[-1, 1]], {n, 100}]

a(n) = vecmax(factor(n^9+1)[,1]); \\ Michel Marcus, Dec 17 2017

cp's OEIS Frontend

A240553 Greatest prime factor of n^10+1.

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

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A223702 Irregular triangle of numbers k such that A002313(n), the n-th prime not congruent to 3 mod 4 is the largest prime factor of k^2 + 1.

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A285283 Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.

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A164314 Largest prime factor of n^2 - 2.

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A240549 Greatest prime factor of n^6+1.

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A285282 Numbers n such that n^2 + 1 is 13-smooth.

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Python

A240550 Greatest prime factor of n^7+1.

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