A223077 -id:A223077 - cp's OEIS frontend

A173951 Positive integers with the property that if the base-3 representation is reversed the result is twice the original number.

32, 104, 320, 968, 2624, 2912, 7808, 8744, 23360, 25376, 26240, 70016, 75920, 78728, 209984, 212576, 227552, 233600, 236192, 629888, 638312, 682448, 700160, 708584, 1889600, 1897376, 1915520, 2047136, 2054912, 2099840, 2117984, 2125760

Offset: 1

John W. Layman, Mar 03 2010

The number of terms of this sequence containing n ternary digits is given by {d(n)}={0,0,0,1,1,1,1,2,2,3,3,5,5,8,8,13,13,21,...} for n=1,2,3,... and thus appears to be essentially the doubling-up of the Fibonacci numbers A103609. For example, 2624 = 10121012(base-3) and 2912 = 10222212(base-3) are the only two terms that have 8 digits when written in base 3, so d(8)=2.
(This conjecture is correct - see A223077. - N. J. A. Sloane, Mar 19 2013)
All terms of sequence A173952, defined by b(1)=32 and, for n>1, b(n)=9*b(n-1)+32, appear to be terms of the above sequence {a(n)}; in fact each term b(n) appears to be the largest term of {a(k)} that has 2n+2 digits when written in base 3.

Cf. A103609, A173952, A223077, A214927.

A223078 Positive integers with the property that if the base-4 representation is reversed the result is three times the original number.

Original entry on oeis.org

75, 315, 1275, 5115, 19275, 20475, 76875, 81915, 307275, 322875, 327675, 1228875, 1290555, 1310715, 4915275, 4934475, 5161275, 5223675, 5242875, 19660875, 19741515, 20644155, 20890875, 20971515, 78643275, 78720075, 78969675, 82575675, 82652475, 83559675

Offset: 1

N. J. A. Sloane, Mar 14 2013

From Robert Israel, Apr 23 2019: (Start)
All terms are divisible by 15.
If x is a term and x < 4^k, then x*(4^k+1) is a term.  In particular the sequence is infinite. (End)

Robert Israel, Table of n, a(n) for n = 1..985

Cf. A173951, A223077, A223079, A214927.

rev4:= proc(n) local L,i;
  L:= convert(n,base,4);
  add(L[-i]*4^(i-1),i=1..nops(L))
end proc:
Res:= NULL:
for d from 2 to 15 do
  d1:= ceil(d/2); d2:= d-d1;
  for a from 4^(d1-1) to 4^d1/3 do
     b:= rev4(a)/3 mod 4^d2;
     x:= 4^d2*a+b;
     if rev4(x) = 3*x then Res:= Res, x; fi
od od:
Res; # Robert Israel, Apr 23 2019

Select[Range[84*10^6],3#==FromDigits[Reverse[IntegerDigits[#,4]],4]&] (* Harvey P. Dale, Mar 03 2018 *)

More terms from Alois P. Heinz, Mar 14 2013

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A173951 Positive integers with the property that if the base-3 representation is reversed the result is twice the original number.

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A223078 Positive integers with the property that if the base-4 representation is reversed the result is three times the original number.

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