A096172 -id:A096172 - cp's OEIS frontend

A081256 Greatest prime factor of n^3 + 1.

2, 3, 7, 13, 7, 31, 43, 19, 73, 13, 37, 19, 157, 61, 211, 241, 13, 307, 7, 127, 421, 463, 13, 79, 601, 31, 37, 757, 271, 67, 19, 331, 151, 1123, 397, 97, 43, 67, 1483, 223, 547, 1723, 139, 631, 283, 109, 103, 61, 181, 43, 2551, 379, 919, 409, 2971, 79, 103, 3307, 163

Offset: 1

Jan Fricke, Mar 14 2003

Record values appear to match the terms of A002383 for n>1. - Bill McEachen, Oct 18 2023

R. J. Mathar and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 5000 terms from R. J. Mathar)
J. Buchmann, K. Győry, M. Mignotte, and N. Tzanakis, Lower bounds for P(x^3+k), an elementary approach, Publ. Math. Debrecen, Vol. 38, No. 1-2 (1991), pp. 145-163.

Cf. A081257, A081258, A096173, A096174.

Cf. A006530, A001093, A002383, A014442, A096172.

A081256 := proc(n)
    A006530(n^3+1) ;
end proc:
seq(A081256(n),n=1..20) ; # R. J. Mathar, Feb 13 2014

Table[Max[Transpose[FactorInteger[n^3 + 1]][[1]]], {n, 25}]

a(n)=my(f=factor(n^3+1)); f[#f~,1] \\ Charles R Greathouse IV, Mar 08 2017

A081256(n)=vecmax(factor(n^3+1)[,1]) \\ It seems slightly slower to get the last element using ...[-1..-1][1]. - M. F. Hasler, Jun 15 2018

a(n) = A006530(A001093(n)). - M. F. Hasler, Jun 13 2018

a(n) >= 31 for n >= 70 (Buchmann et al., 1991). - Amiram Eldar, Oct 25 2024

More terms from Harvey P. Dale, Mar 22 2003

More terms from Hugo Pfoertner, Jun 20 2004

A240548 Greatest prime factor of n^5 + 1.

Original entry on oeis.org

2, 11, 61, 41, 521, 101, 191, 331, 1181, 9091, 13421, 19141, 2411, 101, 1531, 61681, 101, 9041, 2251, 152381, 185641, 224071, 211, 5791, 9161, 1021, 271, 53951, 401, 71261, 21821, 4051, 1151041, 259631, 132631, 6781, 1824841, 2031671, 41011, 20641, 4111, 23201

Offset: 1

T. D. Noe, Apr 07 2014

nonn

			a(2) = 11 because 2^5 + 1 = 33 = 3 * 11.
a(3) = 61 because 3^5 + 1 = 244 = 2^2 * 61.
a(4) = 41 because 4^5 + 1 = 1025 = 5^2 * 41.
a(2272) = 2273 because 2272^5 + 1 = 11^2 * 311 * 491 * 1171 * 1231 * 2273.

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

Cf. A002561, A014442, A081256, A096172, A240549-A240553.

Table[FactorInteger[n^5 + 1][[-1, 1]], {n, 100}]

A240553 Greatest prime factor of n^10+1.

Original entry on oeis.org

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}]

A096169 Odd n such that (n^4+1)/2 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 21, 23, 29, 35, 39, 57, 61, 65, 71, 73, 81, 103, 105, 113, 115, 119, 129, 153, 165, 169, 171, 199, 203, 205, 251, 259, 267, 275, 309, 313, 317, 333, 337, 339, 353, 363, 403, 405, 415, 419, 431, 445, 449, 453, 455, 463, 471, 477, 479, 487

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

			a(1)=3 because (3^4+1)/2=82/2=41 is prime.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000068 n^4+1 is prime, A037896 primes of the form n^4+1, A096170 primes of the form (n^4+1)/2, A096171 n^4+1 is an odd semiprime, A096172 largest prime factor of n^4+1.

[ n: n in [0..2500] | IsPrime((n^4+1) div 2) ]; // Vincenzo Librandi, Apr 15 2011

Select[Range[1,501,2],PrimeQ[(#^4+1)/2]&] (* Harvey P. Dale, Jun 04 2011 *)

A096171 Numbers k such that k^4+1 is an odd semiprime.

Original entry on oeis.org

8, 10, 12, 14, 18, 22, 26, 30, 32, 36, 38, 40, 42, 50, 52, 58, 62, 68, 72, 78, 84, 86, 92, 94, 98, 100, 102, 108, 112, 114, 116, 120, 122, 124, 128, 130, 138, 146, 148, 152, 158, 162, 166, 170, 172, 176, 184, 186, 200, 212, 214, 216, 218, 222, 224, 226, 234, 250, 252

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

			a(1)=8 because 8^4 + 1 = 4097 = 17*241;
a(2)=10: 10^4 + 1 = 10001 = 73*137.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000068 (n^4+1 is prime), A037896 (primes of the form k^4+1), A096169 ((n^4+1)/2 is prime), A069170 (primes of the form (k^4+1)/2), A096172 (largest prime factor of n^4+1), A046388.

Select[Range[2,300,2],PrimeOmega[#^4+1]==2&] (* Harvey P. Dale, Dec 25 2021 *)

isA096171(n) = {local(m);m=n^4+1;(m%2==1)&&(bigomega(m)==2)} \\ Michael B. Porter, Feb 02 2010

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}]

A096170 Primes of the form (k^4 + 1)/2.

Original entry on oeis.org

41, 313, 1201, 7321, 14281, 41761, 97241, 139921, 353641, 750313, 1156721, 5278001, 6922921, 8925313, 12705841, 14199121, 21523361, 56275441, 60775313, 81523681, 87450313, 100266961, 138461441, 273990641, 370600313, 407865361

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

Note that k must be odd. Terms of primitive Pythagorean triples: (k^2, (k^4-1)/2, (k^4+1)/2).

			a(1)=41 because (3^4 + 1)/2 = 82/2 = 41 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A096169 (n^4+1)/2 is prime, A000068 n^4+1 is prime, A037896 primes of the form n^4+1, A096171 n^4+1 is an odd semiprime, A096172 largest prime factor of n^4+1.

[ a: n in [0..2500] | IsPrime(a) where a is ((n^4+1) div 2) ]; // Vincenzo Librandi, Apr 15 2011

Select[(Range[200]^4+1)/2,PrimeQ] (* Harvey P. Dale, Mar 09 2013 *)

list(lim)=my(v=List(),t); forstep(n=3,sqrtnint(lim\1*2-1,4),2, if(isprime(t=(n^4+1)/2), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

Name edited by Zak Seidov, Apr 14 2011

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

A081256 Greatest prime factor of n^3 + 1.

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A240548 Greatest prime factor of n^5 + 1.

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A240553 Greatest prime factor of n^10+1.

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A096169 Odd n such that (n^4+1)/2 is prime.

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A096171 Numbers k such that k^4+1 is an odd semiprime.

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PARI

A240549 Greatest prime factor of n^6+1.

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Mathematica

A096170 Primes of the form (k^4 + 1)/2.

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Magma

Mathematica

PARI

Extensions

A240550 Greatest prime factor of n^7+1.

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