A093074 -id:A093074 - cp's OEIS frontend

A007531 a(n) = n(n-1)(n-2) (or n!/(n-3)!).

0, 0, 0, 6, 24, 60, 120, 210, 336, 504, 720, 990, 1320, 1716, 2184, 2730, 3360, 4080, 4896, 5814, 6840, 7980, 9240, 10626, 12144, 13800, 15600, 17550, 19656, 21924, 24360, 26970, 29760, 32736, 35904, 39270, 42840, 46620, 50616, 54834, 59280, 63960, 68880

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Ed Pegg Jr conjectures that n^3 - n = k! has a solution if and only if n is 2, 3, 5 or 9 (when k is 3, 4, 5 and 6).
Three-dimensional promic (or oblong) numbers, cf. A002378. - Alexandre Wajnberg, Dec 29 2005
Doubled first differences of tritriangular numbers A050534(n) = (1/8)n(n + 1)(n - 1)(n - 2). a(n) = 2*(A050534(n+1) - A050534(n)). - Alexander Adamchuk, Apr 11 2006
If Y is a 4-subset of an n-set X then, for n >= 6, a(n-4) is the number of (n-5)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007
Convolution of A005843 with A008585. - Reinhard Zumkeller, Mar 07 2009
a(n) = A000578(n) - A000567(n). - Reinhard Zumkeller, Sep 18 2009
For n > 3: a(n) = A173333(n, n-3). - Reinhard Zumkeller, Feb 19 2010
Let H be the n X n Hilbert matrix H(i, j) = 1/(i+j-1) for 1 <= i, j <= n. Let B be the inverse matrix of H. The sum of the elements in row 2 of B equals (-1)^n a(n+1). - T. D. Noe, May 01 2011
a(n) equals 2^(n-1) times the coefficient of log(3) in 2F1(n-2, n-2, n, -2). - John M. Campbell, Jul 16 2011
For n > 2 a(n) = 1/(Integral_{x = 0..Pi/2} (sin(x))^5*(cos(x))^(2*n-5)). - Francesco Daddi, Aug 02 2011
a(n) is the number of functions f:[3] -> [n] that are injective since there are n choices for f(1), (n-1) choices for f(2), and (n-2) choices for f(3). Also, a(n+1) is the number of functions f:[3] -> [n] that are width-2 restricted (that is, the pre-image under f of any element in [n] is of size 2 or less). See "Width-restricted finite functions" link below. - Dennis P. Walsh, Mar 01 2012
This sequence is produced by three consecutive triangular numbers t(n-1), t(n-2) and t(n-3) in the expression 2*t(n-1)*(t(n-2)-t(n-3)) for n = 0, 1, 2, ... - J. M. Bergot, May 14 2012
For n > 2: A020639(a(n)) = 2; A006530(a(n)) = A093074(n-1). - Reinhard Zumkeller, Jul 04 2012
Number of contact points between equal spheres arranged in a tetrahedron with n - 1 spheres in each edge. - Ignacio Larrosa Cañestro, Jan 07 2013
Also for n >= 3, area of Pythagorean triangle in which one side differs from hypotenuse by two units. Consider any Pythagorean triple (2n, n^2-1, n^2+1) where n > 1. The area of such a Pythagorean triangle is n(n^2-1). For n = 2, 3, 4,.. the areas are 6, 24, 60, .... which are the given terms of the series. - Jayanta Basu, Apr 11 2013
Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices (chromatic polynomial) of the complete graph K_3. - Tom Copeland, Apr 05 2014
Starting with 6, 24, 60, 120, ..., a(n) is the number of permutations of length n>=3 avoiding the partially ordered pattern (POP) {1>2} of length 5. That is, the number of length n permutations having no subsequences of length 5 in which the first element is larger than the second element. - Sergey Kitaev, Dec 11 2020
For integer m and positive integer r >= 2, the polynomial a(n) + a(n + m) + a(n + 2*m) + ... + a(n + r*m) in n has its zeros on the vertical line Re(n) = (2 - r*m)/2 in the complex plane. - Peter Bala, Jun 02 2024

R. K. Guy, Unsolved Problems in Theory of Numbers, Section D25.
L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 40.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. D. Birkhoff, A determinant formula for the number of ways of coloring a map, Ann. Math., 14:42-4. See 1st polynomial p. 5.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Milan Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Michael Penn, A natural nested root, YouTube video, 2022.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014-2015.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv:1508.07894 [math.NT], 2015.
Dennis Walsh, Width-restricted finite functions
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

binomial(n, k): A161680 (k = 2), A000332 (k = 4), A000389 (k = 5), A000579 (k = 6), A000580 (k = 7), A000581 (k = 8), A000582 (k = 9).
Cf. A002378, A005563, A084939, A084940, A084941, A084942, A084943, A084944.
Cf. A028896.

a007531 n = product [n-2..n]  -- Reinhard Zumkeller, Jul 04 2012

[n*(n-1)*(n-2): n in [0..40]]; // Vincenzo Librandi, May 02 2011

[seq(6*binomial(n,3),n=0..41)]; # Zerinvary Lajos, Nov 24 2006

Table[n^3 - 3n^2 + 2n, {n, 0, 42}]
Table[FactorialPower[n, 3], {n, 0, 42}] (* Arkadiusz Wesolowski, Oct 29 2012 *)

PARI
```
a(n)=n*(n-1)*(n-2)
```

[n*(n-1)*(n-2) for n in range(40)] # G. C. Greubel, Feb 11 2019

a(n) = 6*A000292(n-2).
a(n) = Sum_{i=1..n} polygorial(3,i) where polygorial(3,i) = A028896(i-1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6, n > 2. - Zak Seidov, Feb 09 2006
G.f.: 6*x^2/(1-x)^4.
a(-n) = -a(n+2).
1/6 + 3/24 + 5/60 + ... = Sum_{k>=1} (2*k-1)/(k*(k+1)*(k+2)) = 3/4. [Jolley Eq. 213]
a(n+1) = n^3 - n. - Mohammad K. Azarian, Jul 26 2007
E.g.f.: x^3*exp(x). - Geoffrey Critzer, Feb 08 2009
If the first 0 is eliminated, a(n) = floor(n^5/(n^2+1)). - Gary Detlefs, Feb 11 2010
1/6 + 1/24 + 1/60 + ... = Sum_{n>=1} 1/(n*(n+1)*(n+2)) = 1/4. - Mohammad K. Azarian, Dec 29 2010
a(0) = 0, a(n) = a(n-1) + 3*(n-1)*(n-2). - Jean-François Alcover, Jan 08 2013
(a(n+1) - a(n))/6 = A000217(n-2) for n > 0. - J. M. Bergot, Jul 30 2013
Partial sums of A028896. - R. J. Mathar, Aug 28 2014
1/6 + 1/24 + 1/60 + ... + 1/(n*(n+1)*(n+2)) = n*(n+3)/(4*(n+1)*(n+2)). - Christina Steffan, Jul 20 2015
a(n+2)^2 = A005563(n)^3 + A005563(n)^2. - Bruno Berselli, May 03 2018
a(n)*a(n+1) + A000096(n-3)^2 = m^2 (a perfect square), m = ((a(n)+a(n+1))/2)-n. - Ezhilarasu Velayutham, May 21 2019
Sum_{n>=3} (-1)^(n+1)/a(n) = 2*log(2) - 5/4. - Amiram Eldar, Jul 02 2020
For n >= 3, (a(n) + (a(n) + (a(n) + ...)^(1/3))^(1/3))^(1/3) = n - 1. - Paolo Xausa, Apr 09 2022

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

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)).

A093075 Indicating the neighbor with a greater greatest prime factor.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, -1, 1, 1, 0, 1, 0, -1, -1, 1, 0, 1, 0, -1, 1, 1, 0, -1, 1, 0, -1, 1, 0, 1, 0, -1, 1, 0, -1, 1, 0, -1, -1, 1, 0, 1, 0, -1, 1, 1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 1, 1, 0, 1, 0, -1, -1, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, -1, -1, 0, -1, 1, 0, -1, 1, 1, 0, -1, 1, 0, -1, 1, 0, -1

Offset: 1

Reinhard Zumkeller, Mar 18 2004

sign

A093074(n)=A006530(n+a(n)); A006530(n)<=A006530(n+a(n)).

A103359 T(n,k) = Max{p: prime p divides m where n-k<=m<=n+k}, triangle read by rows: 0<=k

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 2, 5, 5, 7, 5, 5, 7, 7, 7, 3, 7, 7, 7, 7, 11, 7, 7, 7, 7, 11, 11, 13, 2, 7, 7, 11, 11, 13, 13, 13, 3, 5, 11, 11, 13, 13, 13, 13, 17, 5, 11, 11, 13, 13, 13, 13, 17, 17, 19, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 3, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 13, 13

Offset: 1

Reinhard Zumkeller, Mar 20 2005

T(n,0) = A006530(n); T(n,1) = A093074(n) for n>1;

T(n,n-1) = A060265(n) for n>1.

			Triangle begins:
1;
2, 3;
3, 3, 5;
2, 5, 5, 7;
5, 5, 7, 7, 7;
...

T[n_, k_] := FactorInteger[#][[-1, 1]]& /@ Range[n-k, n+k] // Max;
Table[T[n, k], {n, 1, 13}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

A125604 Minimum of the largest prime factors of a number and its two neighbors.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 5, 2, 2, 2, 3, 3, 5, 5, 7, 3, 3, 3, 3, 3, 3, 5, 5, 2, 2, 2, 7, 3, 3, 3, 13, 5, 5, 5, 7, 7, 5, 5, 5, 3, 3, 3, 5, 5, 13, 3, 3, 3, 7, 7, 19, 5, 5, 5, 7, 2, 2, 2, 11, 11, 17, 7, 7, 3, 3, 3, 5, 5, 5, 11, 11, 5, 3, 3, 3, 7, 7, 7, 17, 11, 11, 5, 5, 5, 13, 23, 19, 3, 3, 3, 7, 5, 5

Offset: 3

Carlos Alves, Nov 27 2006

			a(93) = min{lpf(92),lpf(93),lpf(94)} = min{23,31,47} = 23.

Robert Israel, Table of n, a(n) for n = 3..10000

Cf. A006530, A093074.

LPF:= map(t -> max(numtheory:-factorset(t)), [$2..101]):
[seq](min(LPF[i..i+2]),i=1..nops(LPF)-2); # Robert Israel, Jun 16 2025

LPF = FactorInteger[ # ][[ -1, 1]] &; Map[Min[{LPF[ # - 1], LPF[ # ], LPF[ # + 1]}] &, Range[3, 200]]
Min/@Partition[Table[FactorInteger[n][[-1,1]],{n,110}],3,1] (* Harvey P. Dale, May 25 2015 *)

a(n) = my(lpf(k)=vecmax(factor(k)[, 1])); vecmin([lpf(n-1), lpf(n), lpf(n+1)]); \\ Ruud H.G. van Tol, Aug 15 2024

from sympy import primefactors
def a(n): return min(map(lambda n: primefactors(n)[-1], range(n-1,n+2))) # David Radcliffe, Jun 16 2025

a(n) = min{lpf(n-1),lpf(n),lpf(n+1)}, where lpf is the largest prime factor: lpf(k) = A006530(k).

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

A007531 a(n) = n*(n-1)*(n-2) (or n!/(n-3)!).

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

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

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Mathematica

PARI

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A093075 Indicating the neighbor with a greater greatest prime factor.

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A103359 T(n,k) = Max{p: prime p divides m where n-k<=m<=n+k}, triangle read by rows: 0<=k

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Mathematica

A125604 Minimum of the largest prime factors of a number and its two neighbors.

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Programs

Maple

Mathematica

PARI

Python

Formula

A321069 Greatest prime factor of n^3+2.

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Author

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Links

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Programs

Magma

Mathematica

PARI

A007531 a(n) = n(n-1)(n-2) (or n!/(n-3)!).

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