A097905 Triangle where T(n,m) = largest divisor of n! coprime to m.

1, 2, 1, 6, 3, 2, 24, 3, 8, 3, 120, 15, 40, 15, 24, 720, 45, 80, 45, 144, 5, 5040, 315, 560, 315, 1008, 35, 720, 40320, 315, 4480, 315, 8064, 35, 5760, 315, 362880, 2835, 4480, 2835, 72576, 35, 51840, 2835, 4480, 3628800, 14175, 44800, 14175, 145152, 175, 518400, 14175, 44800, 567

Offset

1,2

Comments

Right edge of triangle is sequence A095996.

Links

Table of n, a(n) for n=1..55.

Formula

T(n,m) = Sum_{d|n!} sigma(n!/d)*mu(d)*gcd(d,m). - Ridouane Oudra, Jun 13 2026

Example

T(6,3) = 80 because 80 is largest divisor of 720 which is coprime to 3.
Triangle begins:
       1;
       2,    1;
       6,    3,    2;
      24,    3,    8,    3;
     120,   15,   40,   15,    24;
     720,   45,   80,   45,   144,  5;
    5040,  315,  560,  315,  1008, 35,   720;
   40320,  315, 4480,  315,  8064, 35,  5760,  315;
  362880, 2835, 4480, 2835, 72576, 35, 51840, 2835, 4480;
  ...

Maple

seq(seq(denom(m^n/n!), m=1..n), n=1..11); # Vladeta Jovovic, May 03 2005

Mathematica

Flatten[Table[Table[Select[Divisors[n! ], GCD[ #, m]==1&][[ -1]], {m, 1, n}], {n, 1, 10}]] (* Stefan Steinerberger, Nov 09 2007 *)

Prog

(Python)
from math import factorial, isqrt, comb
from sympy import factorint
from sympy.ntheory.factor_ import digits
def A097905_T(n, m):
    k = factorial(n)
    for p, e in factorint(m).items():
        k //= p**min(e*n, (n-sum(digits(n, p)[1:]))//(p-1))
    return k
def A097905(m):
    a = (n:=isqrt(k:=m<<1))+(k>n*(n+1))
    return A097905_T(a, m-comb(a, 2)) # Chai Wah Wu, Jun 12 2026

Crossrefs

Cf. A095996, A000142, A008683, A000203.

Sequence in context: A120435 A125901 A094307 * A094310 A165908 A121281

Adjacent sequences: A097902 A097903 A097904 * A097906 A097907 A097908

Keyword

nonn,tabl,changed

Author

Leroy Quet, Sep 04 2004, Apr 10 2007

Extensions

More terms from Stefan Steinerberger, Nov 09 2007

Status

approved