A074399 -id:A074399 - cp's OEIS frontend

A085153 All prime factors of n and n+1 are <= 7. (Related to the abc conjecture.)

1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 125, 224, 2400, 4374

Offset: 1

Benoit Cloitre, Jun 21 2003

The ABC conjecture would imply that if the prime factors of A, B, C are prescribed in advance, then there is only a finite number of solutions to the equation A + B = C with gcd(A,B,C)=1 (indeed it would bound C to be no more than "roughly" the product of those primes). So in particular there ought to be only finitely many pairs of adjacent integers whose prime factors are limited to {2, 3, 5, 7} (D. Rusin).
This sequence is complete by a theorem of Stormer. See A002071. - T. D. Noe, Mar 03 2008
This is the 4th row of the table A138180. It has 23=A002071(4)=A145604(1)+...+ A145604(4) terms and ends with A002072(4)=4374. It is the union of all terms in rows 1 through 4 of the table A145605. It is a subsequence of A252494 and contains A085152 as a subsequence. - M. F. Hasler, Jan 16 2015
Equivalently, this is the sequence of numbers for which A074399(n) <= 7, or A252489(n) <= 4.

Abderrahmane Nitaj, The ABC conjecture homepage
OEIS Index entries for sequences related to the abc conjecture

Cf. A085152, A002473, A086247.

Select[Range[10000], FactorInteger[ # (# + 1)][[ -1,1]] <= 7 &] (* T. D. Noe, Mar 03 2008 *)

for(n=1,9e6,vecmax(factor(n++)[,1])<8 && vecmax(factor(n--+(n<2))[,1])<8 && print1(n",")) \\ M. F. Hasler, Jan 16 2015

Edited by Dean Hickerson, Jun 30 2003

A076605 Largest prime divisor of n^2 - 1.

Original entry on oeis.org

3, 2, 5, 3, 7, 3, 7, 5, 11, 5, 13, 7, 13, 7, 17, 3, 19, 5, 19, 11, 23, 11, 23, 13, 5, 13, 29, 7, 31, 5, 31, 17, 11, 17, 37, 19, 37, 19, 41, 7, 43, 11, 43, 23, 47, 23, 47, 5, 17, 13, 53, 13, 53, 7, 19, 29, 59, 29, 61, 31, 61, 31, 13, 11, 67, 17, 67, 17, 71, 7

Offset: 2

Jon Perry, Oct 21 2002

nonn

Also the largest prime that divides either n-1 or n+1.

Størmer shows that a(n) tends to infinity with n. Schinzel shows that lim inf a(n)/log log n >= 2 and, using lower bounds for linear forms of logarithms, this inequality can be generalized for general quadratic polynomials, with 2 replaced by 4/7 for irreducible ones and 2/7 for reducible ones. - Tomohiro Yamada, Apr 15 2017

			n=11: the largest prime factor of 10 and 12 is 5, therefore a(11) = 5.

K. Mahler, "Uber den grossten Primteiler spezieller Polynome zweiten Grades", Arch. Math. Naturvid. B.41, 1935, pp. 3 - 26.

T. D. Noe, Table of n, a(n) for n = 2..10000
D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57-79.
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arith. 13 (1967), 177--236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter udgivne af Videnskabsselskabet i Christiania: Mathematisk-naturvidenskabelig Klasse, 2 (1897), 48 pp.

Cf. A006530, A037464, A074399 (bisections).
Cf. A175607.
Cf. A014442 (largest prime divisor of n^2 + 1). - Tomohiro Yamada, Apr 15 2017

Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 2, 80}]

for (n=3,100, print1(","max(factor(n-1)[,1][length(factor(n-1)[,1])],factor(n+1)[,1][length(factor(n+1)[,1])])))

A069902 Largest prime factor of n(n+1)/2, the n-th triangular number.

Original entry on oeis.org

1, 3, 3, 5, 5, 7, 7, 3, 5, 11, 11, 13, 13, 7, 5, 17, 17, 19, 19, 7, 11, 23, 23, 5, 13, 13, 7, 29, 29, 31, 31, 11, 17, 17, 7, 37, 37, 19, 13, 41, 41, 43, 43, 11, 23, 47, 47, 7, 7, 17, 17, 53, 53, 11, 11, 19, 29, 59, 59, 61, 61, 31, 7, 13, 13, 67

Offset: 1

Reinhard Zumkeller, Apr 10 2002

Essentially the same as A074399, which has many comments, references and links.

			A000217(9) = 9*(9+1)/2 = 45 = 3*3*5, therefore a(9) = 5.

Related properties of triangular numbers: A069901, A069903, A069904.

Cf. A000217, A006530, A074399.

PrimeFactors[n_]:=Flatten[Table[ #[[1]],{1}]&/@FactorInteger[n]]; Table[PrimeFactors[n*(n-1)/2][[ -1]],{n,2,6!}] (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
(* Second program: *)
Array[FactorInteger[PolygonalNumber[#]][[-1, 1]] &, 66] (* Michael De Vlieger, Sep 14 2023 *)

\\ written for a(n), n >= 2
a(n)=vecmax(factor(n*(n+1)/2)[,1]) \\ M. F. Hasler, May 02 2015

a(n) = A006530(A000217(n)).

Edited by Peter Munn, Sep 14 2023

A093074 Greatest prime factor of n and its direct neighbors.

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 7, 7, 5, 11, 11, 13, 13, 13, 7, 17, 17, 19, 19, 19, 11, 23, 23, 23, 13, 13, 13, 29, 29, 31, 31, 31, 17, 17, 17, 37, 37, 37, 19, 41, 41, 43, 43, 43, 23, 47, 47, 47, 7, 17, 17, 53, 53, 53, 11, 19, 29, 59, 59, 61, 61, 61, 31, 13, 13, 67, 67, 67, 23, 71, 71, 73

Offset: 1

Reinhard Zumkeller, Mar 18 2004

nonn

a(n) = A006530(n + A093075(n));
a(n) = max{A006530(n-1), A006530(n), A006530(n+1)}, n>1;
a(n) = A006530(A007531(n+1)), n>1;
for all primes p>2: a(p)=a(p-1)=p and if p is not the lesser member of a twin prime pair, then also a(p+1)=p;
(n,n+2) is a twin prime pair iff a(n-1)=a(n)=n and a(n+1)=a(n+2)=a(n+3)=n+2.

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

Cf. A074399, A190136.

a093074 1 = 2
a093074 n = maximum $ map a006530 [n-1..n+1]
-- Reinhard Zumkeller, Jul 04 2012

a(n)=my(p=precprime(n+1));if(p>n-2,p,vecmax(apply(n->vecmax(factor(n)[,1]),[n-1,n,n+1]))) \\ Charles R Greathouse IV, Feb 19 2013

a(n) > 47 if n > 212381. - Charles R Greathouse IV, Feb 19 2013

A037464 Bisection of A076605.

Original entry on oeis.org

3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 5, 29, 31, 31, 11, 37, 37, 41, 43, 43, 47, 47, 17, 53, 53, 19, 59, 61, 61, 13, 67, 67, 71, 73, 73, 11, 79, 79, 83, 83, 29, 89, 89, 31, 31, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 23, 17, 17, 41, 41, 127, 127, 131, 131, 19

Offset: 1

N. J. A. Sloane, Nov 29 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A076605, A074399.

Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 2, 150, 2}]

a(n) = my(p = factor(4*n^2-1)[, 1]); p[#p]; \\ Amiram Eldar, Nov 03 2024

a(n) = max(A006530(2n-1), A006530(2n+1)).

a(n) = A076605(2*n). - Amiram Eldar, Nov 03 2024

Extended by Robert G. Wilson v, Dec 02 2002

A190136 Largest prime factor of n(n+1)(n+2)*(n+3).

Original entry on oeis.org

3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 13, 13, 29, 29, 31, 31, 31, 31, 17, 17, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 17, 17, 53, 53, 53, 53, 19, 29, 59, 59, 61, 61, 61, 61, 31, 13, 67, 67, 67

Offset: 1

Reinhard Zumkeller, May 07 2011

nonn

a(n) > 11 for n > 9;
a(A086801(n)) = A000040(n) for n > 2.
It follows from Størmer's theorem that lim inf a(n) = infinity, and in fact a(n) >> log log n. - Charles R Greathouse IV, Feb 19 2013

			Numbers m <= 10^6 such that a(m) = p:
p=13: 10, 11, 12, 13, 24, 25, 63;
p=17: 14, 15, 32, 33, 48, 49;
p=19: 16, 17, 18, 19, 54, 75, 168;
p=23: 20, 21, 22, 23, 207, 322;
p=29: 26, 27, 55, 114;
p=31: 28, 29, 30, 31, 62, 90, 152, 153, 340, 493, 1518;
p=37: 34, 35, 36, 37, 74, 184, 405;
p=41: 38, 39, 123, 245, 285, 286, 287, 492, 1023, 1517, 1680;
p=43: 40, 41, 42, 43, 84, 85, 169, 258, 341, 342, 558, 1330, 1331, 2106, 5289, 10878;
p=47: 44, 45, 46, 47, 91, 92, 93, 185, 186, 187, 374, 375, 702, 986, 987, 17575;
p=53: 50, 51, 52, 53, 159, 368, 369, 527, 845, 899, 900, 1375;
p=59: 56, 57, 115, 116, 117, 118, 174, 294, 528, 529, 530, 648, 943, 1885, 6783;
p=61: 58, 59, 60, 61, 119, 120, 121, 122, 182, 183, 242, 243, 244, 549, 608, 609, 1034, 1218, 1219, 1767, 1768, 2013, 2254, 2622;
p=67: 64, 65, 66, 67, 132, 133, 735, 1271, 1272, 1273, 2208, 2277, 3885, 4958, 5828, 5829;
p=71: 68, 69, 140, 141, 142, 284, 423, 424, 494, 636, 637, 779, 780, 781, 3477, 3478, 3549, 3550, 4899;
p=73: 70, 71, 72, 73, 143, 144, 145, 219, 363, 510, 728, 729, 803, 1022, 1239, 1679, 2772, 70224;
p=79: 76, 77, 78, 79, 158, 234, 235, 472, 473, 474, 550, 867, 868, 1024, 1104, 1419, 2209, 2448, 2923, 3476, 3869, 4898, 5290, 7502, 46136, 70150;
p=83: 80, 81, 82, 83, 246, 247, 413, 495, 663, 664, 1078, 1159, 1824, 2736, 3483, 4232, 4896, 4897, 7137, 8214, 12614, 36517, 97524;
p=89: 86, 87, 88, 89, 175, 264, 265, 354, 531, 710, 711, 712, 798, 1245, 1332, 2847, 4895, 5073, 6318, 18423, 28302, 29279;
p=97: 94, 95, 96, 97, 288, 289, 483, 580, 581, 582, 774, 873, 1064, 1065, 1455, 2132, 2133, 3007, 3975, 4556, 4557, 6496, 6497, 6887, 7564, 7565, 7566, 13869, 17457.

Paulo Ribenboim, Galimatias Arithmeticae (Chap 11), in 'My Numbers, My Friends', Springer-Verlag 2000 NY, page 345.
J. J. Sylvester, "On arithmetical series", Messenger of Mathematics 21 (1892), pp. 1-19 and 87-120.
M. Faulkner, "On a theorem of Sylvester and Schur", J. London Math. Soc. 41:1 (1966), pp. 107-110.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A006530, A074399, A093074, A193945.

a190136 n = maximum $ map a006530 [n..n+3]

Table[FactorInteger[Times@@(n+Range[0,3])][[-1,1]],{n,70}] (* Harvey P. Dale, Mar 19 2018 *)

gpf(n)=vecmax(factor(n)[,1])
a(n)=my(p=precprime(n+3));if(pCharles R Greathouse IV, Feb 19 2013

a(n) = max{gpf(n), gpf(n+1), gpf(n+2), gpf(n+3)} = gpf(A052762(n+3)) with gpf = A006530, greatest prime factor.

a(n) > 47 for n > 17575. - Charles R Greathouse IV, Feb 19 2013

A252489 Index of the largest prime which divides n*(n+1).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 2, 3, 5, 5, 6, 6, 4, 3, 7, 7, 8, 8, 4, 5, 9, 9, 3, 6, 6, 4, 10, 10, 11, 11, 5, 7, 7, 4, 12, 12, 8, 6, 13, 13, 14, 14, 5, 9, 15, 15, 4, 4, 7, 7, 16, 16, 5, 5, 8, 10, 17, 17, 18, 18, 11, 4, 6, 6, 19, 19, 9, 9, 20, 20, 21, 21, 12, 8, 8, 6

Offset: 1

M. F. Hasler, Jan 16 2015

Yields the row of A145605 in which n appears, and also the first row of A138180 in which n appears.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002071, A145604, A138180, A145605, A002072, A074399, A061395.

A061395:= [1, seq(numtheory:-pi(max(numtheory:-factorset(n))), n=2..101)]:
zip(max,A061395[1..-2],A061395[2..-1]); # Robert Israel, Feb 12 2021

a[n_] := PrimePi[Max[FactorInteger[n][[-1, 1]], FactorInteger[n+1][[-1, 1]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 05 2023 *)

a(n)=primepi(vecmax(factor(n*(n+1))[,1]))

a(n) = pi(A074399(n)), where pi = A000720.

a(n) = max(A061395(n),A061395(n+1)). - Robert Israel, Feb 12 2021

A199423 Greatest prime factor of n and 2*n+1.

Original entry on oeis.org

3, 5, 7, 3, 11, 13, 7, 17, 19, 7, 23, 5, 13, 29, 31, 11, 17, 37, 19, 41, 43, 11, 47, 7, 17, 53, 11, 19, 59, 61, 31, 13, 67, 23, 71, 73, 37, 19, 79, 5, 83, 17, 43, 89, 13, 31, 47, 97, 11, 101, 103, 13, 107, 109, 37, 113, 23, 29, 59, 11, 61, 31, 127, 43, 131, 19

Offset: 1

Franklin T. Adams-Watters, Nov 06 2011

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A006530, A014105, A074399, A093074, A076605.

Table[Max[Flatten[FactorInteger[{n,2 n+1}],1][[All,1]]],{n,70}] (* Harvey P. Dale, Mar 25 2020 *)

gpf(n)=local(ps);if(n<2,n,ps=factor(n)[,1]~;ps[#ps])
vector(80,n,gpf(n*(2*n+1)))

a(n) = A006530(A014105(n)).

A321069 Greatest prime factor of n^3+2.

Original entry on oeis.org

3, 5, 29, 11, 127, 109, 23, 257, 43, 167, 43, 173, 733, 1373, 307, 683, 983, 2917, 2287, 4001, 157, 71, 283, 223, 5209, 47, 127, 3659, 24391, 587, 9931, 113, 433, 6551, 809, 569, 307, 27437, 433, 10667, 439, 239, 1559, 223, 91127, 16223, 4153, 457, 39217, 62501

Offset: 1

Charles R Greathouse IV, Oct 27 2018

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, The largest prime factor of x^3+2, Proceedings of the London Mathematical Society, 82:3 (2000), pp. 554-596.
Christopher Hooley, On the greatest prime factor of a cubic polynomial, Journal für die reine und angewandte Mathematik, 303 (1978), pp. 21-50.
A. J. Irving, The largest prime factor of x^3+2, arXiv:1412.0024 [math.NT], 2014.

Greatest prime factors of polynomials: A006530 (n), A076565 (2n+1), A076566 (3n+3), A076567 (4n+6), A164314 (n^2-2), A076605 (n^2-1), A014442 (n^2+1), A069902 (n^2+n), A074399 (n^2+n), A199423 (2n^2+n), A089619 (2n^2+2n+1), A037464 (4n^2-1), A253254 (9n^2-7n), A093074 (n^3-n), A081257 (n^3-1), A081256 (n^3+1), A321069(n^3+2), A281793 (n^3+n^2+n+1), A281793 (n^4-1), A096172 (n^4+1), A190136 (n^4 + 6n^3 + 11n^2 + 6n), A140538 (2n^4+1), A240548 (n^5+1), A281794 (n^5+n^3+n^2+1), A240549 (n^6+1), A240550 (n^7+1), A240551 (n^8+1), A240552 (n^9+1), A240553 (n^10+1).

[Maximum(PrimeDivisors(n^3 + 2)): n in [1..60]]; // Vincenzo Librandi, Oct 27 2018

Table[FactorInteger[n^3 + 2] [[-1, 1]], {n, 80}] (* Vincenzo Librandi, Oct 27 2018 *)

a(n) = vecmax(factor(n^3+2)[,1]); \\ Michel Marcus, Oct 27 2018

cp's OEIS Frontend

A085153 All prime factors of n and n+1 are <= 7. (Related to the abc conjecture.)

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A076605 Largest prime divisor of n^2 - 1.

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A069902 Largest prime factor of n(n+1)/2, the n-th triangular number.

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A093074 Greatest prime factor of n and its direct neighbors.

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Haskell

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A037464 Bisection of A076605.

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A190136 Largest prime factor of n*(n+1)*(n+2)*(n+3).

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Haskell

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A252489 Index of the largest prime which divides n*(n+1).

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A190136 Largest prime factor of n(n+1)(n+2)*(n+3).