A073095 Numbers k such that the final nonzero digit of k! is the same as the last digit of binomial(2k,k) (in base 10).

5, 12, 26, 31, 35, 51, 56, 60, 136, 152, 157, 177, 182, 252, 257, 275, 280, 287, 300, 305, 312, 627, 632, 650, 655, 662, 675, 680, 687, 751, 756, 760, 786, 811, 886, 902, 907, 927, 932, 1251, 1256, 1260, 1286, 1311, 1377, 1382, 1400, 1405, 1412, 1425

Offset

1,1

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Formula

k such that A008904(k) = binomial(2k, k) reduced (mod 10).

Example

12! = 479001600, binomial(24,12) = 2704156, and the last nonzero digit of 12! is the same as the last digit of binomial(24,12), hence 12 is in the sequence.

Mathematica

Select[Range[1500], Mod[#!/10^IntegerExponent[#!, 10], 10]==Mod[Binomial[2 #, #], 10]&] (* Harvey P. Dale, Sep 13 2022 *)

Prog

(Python)
from math import comb
from functools import reduce
from itertools import count, zip_longest, islice
from sympy.ntheory.factor_ import digits
from sympy.ntheory.modular import crt
def A073095_gen(startvalue=2): # generator of terms >= startvalue
    for n in count(max(startvalue, 2)):
        s, s2 = digits(n, 5)[-1:0:-1], digits(n<<1, 5)[-1:0:-1]
        if reduce(lambda x, y:x*y%10, (((6, 2, 4, 8, 6, 2, 4, 8, 2, 4, 8, 6, 6, 2, 4, 8, 4, 8, 6, 2)[(a<<2)|(i*a&3)] if i*a else (1, 1, 2, 6, 4)[a]) for i, a in enumerate(s)), 6)==crt([2, 5], [0, reduce(lambda x, y:x*y%5, (comb(a, b) for a, b in zip_longest(s2, s, fillvalue=0)))])[0]:
            yield n
A073095_list = list(islice(A073095_gen(), 50)) # Chai Wah Wu, Dec 07 2023

Crossrefs

Sequence in context: A223233 A038254 A223321 * A309323 A364970 A294017

Adjacent sequences: A073092 A073093 A073094 * A073096 A073097 A073098

Keyword

base,easy,nonn

Author

Benoit Cloitre, Aug 18 2002

Status

approved