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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).
3
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
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
KEYWORD
base,easy,nonn
AUTHOR
Benoit Cloitre, Aug 18 2002
STATUS
approved