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A058186 Numbers (written in base 5) which appear the same when written in base 5 and base 10/2.

%I A058186 #11 Aug 02 2025 06:26:26
%S A058186 0,1,2,3,4,20,21,22,23,24,40,41,42,43,44,200,201,202,203,204,220,221,
%T A058186 222,223,224,240,241,242,243,244,400,401,402,403,404,420,421,422,423,
%U A058186 424,440,441,442,443,444,2000,2001,2002,2003,2004,2020,2021,2022,2023
%N A058186 Numbers (written in base 5) which appear the same when written in base 5 and base 10/2.
%C A058186 To represent a number in base b, if a digit exceeds b-1, subtract b and carry 1. In fractional base b/c, subtract b and carry c. The sequence consists of numbers which in base 5 have digits in {0,2,4} except that the unit digit can be any from {0,1,2,3,4}.
%H A058186 Amiram Eldar, <a href="/A058186/b058186.txt">Table of n, a(n) for n = 1..10000</a>
%H A058186 <a href="/index/Ba#base_fractional">Index entries for sequences related to fractional bases</a>.
%F A058186 a(n) = A007091(A058185(n)). - _Amiram Eldar_, Aug 02 2025
%e A058186 20 (10 in decimal) is a term since it is written as 20 both in base 5 and base 10/2.
%e A058186 30 (15 in decimal) is not a term since it is written as 30 in base 5 and 25 in base 10/2.
%t A058186 s[n_] := s[n] = If[n == 0, 0, 10*s[2*Floor[n/10]] + Mod[n, 10]]; f[n_] := FromDigits[IntegerDigits[n, 5]]; q[k_] := s[k] == f[k]; f /@ Select[Range[0, 300], q] (* _Amiram Eldar_, Aug 02 2025 *)
%o A058186 (PARI) s(n) = if(n == 0, 0, 10 * s(n\10 * 2) + n % 10);
%o A058186 f(n) = fromdigits(digits(n, 5));
%o A058186 list(lim) = apply(f, select(x -> s(x) == f(x), vector(lim+1, i, i-1))); \\ _Amiram Eldar_, Aug 02 2025
%Y A058186 Cf. A007091, A024657, A058185.
%K A058186 base,nonn,easy
%O A058186 1,3
%A A058186 _Henry Bottomley_, Nov 17 2000
%E A058186 Offset corrected by _Amiram Eldar_, Aug 02 2025