A382638 Numbers k for which the repeating part with leading 0's of 1/k in decimal is a palindrome and longer than one digit.

1616, 14208, 16160, 17472, 142080, 161600, 174720, 454656, 511488, 838656, 1363968, 1420800, 1578125, 1616000, 1747200, 1818624, 1900992, 4091904, 4265625, 4546560, 4734375, 5114880, 8183808, 8386560, 13639680, 14208000, 15781250, 16160000, 17472000, 18186240, 19009920

Offset: 1

Author

Jean-Marc Rebert, Apr 01 2025

Keywords

• base
• nonn

Examples

			1616 is a term, because 1/1616 = 0.0006188118811881188118811881188118811881... = 0.0006(1881), where the repeating period 1881 is a palindrome longer than a single digit.
511488 is a term, because 1/511488 = 1.955080080080080080080080080080080080... E-6 = 1.955(080) E-6, where the repeating period O80 is a palindrome longer than a single digit.
11 is not a term, because 1/11 = 0.09090909090909090909090909090909090909... = 0,(09), where the repeating period 09 is not a palindrome .
101 is not a term, because 1/101 = 0.0099009900990099009900990099009900990099 = 0,(0099), where the repeating period 0099 is not a palindrome.
Term 4091904 is itself a palindrome. - _Bert Dobbelaere_, Apr 27 2025
		

Crossrefs

Cf. A060284, A060283.

Cf. A382176 (in golden ratio base).

Programs

Mathematica

p[{t_List}]:=t; p[t_List]:={}; p[{, t_List}]:=t; Select[ Range@ 20000, (r = p@ RealDigits[1/#][[1]]; Length@ r > 1 && r == Reverse@ r) &] (* Giovanni Resta, Apr 23 2025 *)

Python

from itertools import count, islice
from sympy import multiplicity, n_order
def A382638_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,k), multiplicity(5,k)
        r = max(m2,m5)
        b, m = 10**r, 10**(t:=n_order(10,c) if (c:=(k>>m2)//5**m5)>1 else 1)-1
        s = str(m*b//k-b//k*m).zfill(t)
        if len(s)>1 and s[:(l:=len(s)+1>>1)]==s[:-l-1:-1]:
            yield k
A382638_list = list(islice(A382638_gen(),4)) # Chai Wah Wu, Apr 22 2025

Extensions

More terms from Bert Dobbelaere, Apr 27 2025