A093888 -id:A093888 - cp's OEIS frontend

A093889 a(n) = n!/A093888(n).

1, 1, 1, 1, 3, 15, 80, 20, 160, 1440, 75, 825, 9900, 128700, 165888, 2488320, 39813120, 597196800, 10749542400, 125411328000, 2508226560000, 52672757760000, 2769091920000, 4901791334400, 117642992025600, 2941074800640000, 76467944816640000, 2064634510049280000, 57809766281379840000, 1676483222160015360000

Offset: 0

Amarnath Murthy, Apr 23 2004

			a(4) = 3 as the largest palindromic divisor of 4! comes from the set {1, 2, 3, 4, 6, 8, 12, 24}. The largest palindrome is this set is 8 so a(4) = 4! / 8 = 3. - _David A. Corneth_, Oct 12 2022

Cf. A000142, A093888.

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

Corrected and extended by Jason Earls, May 07 2004

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

A093030 Largest palindromic divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 5, 11, 6, 1, 7, 5, 8, 1, 9, 1, 5, 7, 22, 1, 8, 5, 2, 9, 7, 1, 6, 1, 8, 33, 2, 7, 9, 1, 2, 3, 8, 1, 7, 1, 44, 9, 2, 1, 8, 7, 5, 3, 4, 1, 9, 55, 8, 3, 2, 1, 6, 1, 2, 9, 8, 5, 66, 1, 4, 3, 7, 1, 9, 1, 2, 5, 4, 77, 6, 1, 8, 9, 2, 1, 7, 5, 2, 3, 88, 1, 9, 7, 4, 3, 2, 5, 8, 1, 7, 99, 5, 101

Offset: 1

Jason Earls, May 07 2004

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

Cf. A093888.

a[n_] := Module[{d = Divisors[n]}, k = Length[d]; While[!PalindromeQ[d[[k]]], k--]; d[[k]]]; Array[a, 101] (* Amiram Eldar, Dec 16 2019 *)

A339508 Number of subsets of {2..n} such that the product of the elements is a decimal palindrome.

Original entry on oeis.org

1, 1, 2, 4, 6, 7, 8, 10, 11, 13, 13, 33, 43, 56, 70, 73, 99, 114, 134, 151, 151, 185, 333, 372, 445, 456, 558, 565, 672, 1031, 1031, 1220, 1518, 1967, 2161, 2176, 2238, 5399, 5543, 6720, 6720, 8857, 10019, 11819, 16882, 16896, 18072, 19299, 21267, 23405, 23405, 24686

Offset: 0

Ilya Gutkovskiy, Dec 07 2020

a(k*10) = a(k*10-1) because all new products end in 0, thus not palindromes. - Michael S. Branicky, Dec 08 2020

			a(9) = 13 subsets: {}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {2, 3}, {2, 4}, {4, 7, 9} and {2, 3, 6, 7}.

David A. Corneth, List of factorizations of the a(n) products for n = 1..51.
David A. Corneth, Lower bounds on a(n) for n = 0..100.
Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes

Cf. A002113, A093888, A339507.

from numpy import prod
from itertools import combinations
def a(n):
    ans = 1  # empty set
    for r in range(1, n):
        for s in combinations(range(2, n+1), r):
            strsp = str(prod(s))
            ans += strsp==strsp[::-1]
    return ans
print([a(n) for n in range(20)])  # Michael S. Branicky, Dec 08 2020

from functools import lru_cache, reduce
from operator import mul
from itertools import combinations
@lru_cache(maxsize=None)
def A339508(n):
    nlist = [i for i in range(2,n) if i % 10 != 0]
    if n == 0 or n == 1:
        return 1
    c = A339508(n-1)
    if n % 10 != 0:
        if str(n) == str(n)[::-1]:
            c += 1
        for i in range(1,len(nlist)+1):
            for d in combinations(nlist,i):
                s = str(reduce(mul,d)*n)
                if s == s[::-1]:
                    c += 1
    return c # Chai Wah Wu, Dec 08 2020

from functools import lru_cache
def cond(p): sp = str(p); return sp == sp[::-1]
@lru_cache(maxsize=None)
def b(n, p):
    if n == 1: return int(cond(p))
    return b(n-1, p) + b(n-1, p*n) if p*n%10 else b(n-1, p)
def a(n): return 1 if n < 2 else b(n, 1)
print([a(n) for n in range(36)]) # Michael S. Branicky, Oct 05 2022

a(23)-a(32) from Michael S. Branicky, Dec 08 2020
a(33)-a(42) from Chai Wah Wu, Dec 08 2020
a(43)-a(48) from Chai Wah Wu, Dec 11 2020
a(49)-a(51) from Michael S. Branicky, Oct 05 2022

A114340 Largest palindromic divisor of n!! (double factorial = A006882(n)).

Original entry on oeis.org

1, 2, 3, 8, 5, 8, 7, 8, 9, 8, 99, 9, 9009, 252, 9009, 252, 9009, 48384, 969969, 48384, 969969, 405504, 969969, 405504, 969969, 405504, 969969, 525525, 969969, 525525, 79833897, 525525, 133464331, 595595, 5273993725, 595595

Offset: 1

Giovanni Resta, Feb 07 2006

			a(16)=252 since 252 is the largest palindromic factor of 16!!= 10321920=252* 40960.

Cf. A006882, A093030, A093888.

palQ[n_]:= n == FromDigits@Reverse@IntegerDigits@n; Array[Max[Select[Divisors[ #!! ], palQ]] &, 40]

a(n) = A093030(A006882(n)). - Michel Marcus, Oct 13 2022

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A093889 a(n) = n!/A093888(n).

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A093030 Largest palindromic divisor of n.

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A339508 Number of subsets of {2..n} such that the product of the elements is a decimal palindrome.

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A114340 Largest palindromic divisor of n!! (double factorial = A006882(n)).

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