A382638 - OEIS

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

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OFFSET

1,1

LINKS

Table of n, a(n) for n=1..31.

EXAMPLE

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

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 *)

PROG

(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