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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.

%I A382638 #42 Apr 27 2025 09:07:54
%S A382638 1616,14208,16160,17472,142080,161600,174720,454656,511488,838656,
%T A382638 1363968,1420800,1578125,1616000,1747200,1818624,1900992,4091904,
%U A382638 4265625,4546560,4734375,5114880,8183808,8386560,13639680,14208000,15781250,16160000,17472000,18186240,19009920
%N 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.
%e A382638 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.
%e A382638 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.
%e A382638 11 is not a term, because 1/11 = 0.09090909090909090909090909090909090909... = 0,(09), where the repeating period 09 is not a palindrome .
%e A382638 101 is not a term, because 1/101 = 0.0099009900990099009900990099009900990099 = 0,(0099), where the repeating period 0099 is not a palindrome.
%e A382638 Term 4091904 is itself a palindrome. - _Bert Dobbelaere_, Apr 27 2025
%t A382638 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 *)
%o A382638 (Python)
%o A382638 from itertools import count, islice
%o A382638 from sympy import multiplicity, n_order
%o A382638 def A382638_gen(startvalue=1): # generator of terms >= startvalue
%o A382638     for k in count(max(startvalue,1)):
%o A382638         m2, m5 = multiplicity(2,k), multiplicity(5,k)
%o A382638         r = max(m2,m5)
%o A382638         b, m = 10**r, 10**(t:=n_order(10,c) if (c:=(k>>m2)//5**m5)>1 else 1)-1
%o A382638         s = str(m*b//k-b//k*m).zfill(t)
%o A382638         if len(s)>1 and s[:(l:=len(s)+1>>1)]==s[:-l-1:-1]:
%o A382638             yield k
%o A382638 A382638_list = list(islice(A382638_gen(),4)) # _Chai Wah Wu_, Apr 22 2025
%Y A382638 Cf. A060284, A060283.
%Y A382638 Cf. A382176 (in golden ratio base).
%K A382638 base,nonn
%O A382638 1,1
%A A382638 _Jean-Marc Rebert_, Apr 01 2025
%E A382638 More terms from _Bert Dobbelaere_, Apr 27 2025