A093888 Largest palindromic divisor of n!.

1, 1, 2, 6, 8, 8, 9, 252, 252, 252, 48384, 48384, 48384, 48384, 525525, 525525, 525525, 595595, 595595, 969969, 969969, 969969, 405909504, 5273993725, 5273993725, 5273993725, 5273993725, 5273993725, 5273993725, 5273993725, 5273993725, 290826628092, 290826628092, 290826628092

Offset: 0

Author

Amarnath Murthy, Apr 23 2004

Keywords

• base
• nonn

Comments

As positive palindromes do not end in 0 terms are not a multiple of 10. - David A. Corneth, Oct 07 2022

Examples

			a(8) = 252 as 252 is the largest palindrome that divides 8! = 40320.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 0..60 (terms 1..40 from _David A. Corneth_)
• David A. Corneth, Lower bounds for a(0)..a(100)

Crossrefs

Cf. A000142, A002113, A093030, A093889.

Programs

Mathematica

Table[SelectFirst[Reverse[Divisors[n!]],PalindromeQ],{n,30}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 19 2020 *)

PARI

a(n) = { res = 1; my(d = divisors(n! >> val(n, 2))); forstep(i = #d, 1, -1, if(ispal(d[i]), res = d[i]; break; ) ); d = divisors(n! / 5^val(n, 5)); forstep(i = #d, 1, -1, if(d[i] < res, return(res); ); if(ispal(d[i]), res = d[i]; break; ) ); res }
ispal(n) = my(d = digits(n)); d == Vecrev(d)
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Oct 07 2022

Python

from sympy import divisors, factorial, multiplicity
def ispal(n): s = str(n); return s == s[::-1]
def b(n, k): f = factorial(n); return f//k**multiplicity(k, f)
def a(n):
    m2 = max(d for d in divisors(b(n, 2), generator=True) if ispal(d))
    m5 = max(d for d in divisors(b(n, 5), generator=True) if ispal(d))
    return max(m2, m5)
print([a(n) for n in range(34)]) # Michael S. Branicky, Oct 12 2022 after David A. Corneth

Formula

a(n) = A093030(A000142(n)). - Michel Marcus, Oct 12 2022

Extensions

More terms from Jason Earls, May 07 2004

a(0) and more terms from David A. Corneth, Oct 07 2022