A169828 - OEIS

%I #20 Aug 26 2020 11:06:18

%S 0,1,1002,1003,16495194,16495195,988496761674,988496761675,

%T 26688254956950,26688254956951,34018805387982,34018805387983,

%U 18069028755380465688,18069028755380465689,38281545036919307556,38281545036919307557

%N Numbers n such that the base 2 expansion of n ends with the base 3 expansion of n.

%C The entries appear to occur in pairs, (6k, 6k+1). - _Zak Seidov_, May 31 2010. Proof from _Max Alekseyev_, May 31 2010: If m is an even term of A169828, then it is also divisible by 3, i.e., m=6k, and m+1 is also a term of A169828. On the other hand, if m is an odd term of A169828, then m == 1 (mod 3) and thus m=6k+1, implying that m-1 is also a term of A169828.

%H Max Alekseyev, <a href="/A169828/b169828.txt">Table of n, a(n) for n = 1..30</a>

%H Erich Friedman, <a href="https://erich-friedman.github.io/numbers.html">What's Special About This Number?</a> (See entries 1002, 1003.)

%e This is Maple:

%e convert(1002,base,2);

%e [0, 1, 0, 1, 0, 1, 1, 1, 1, 1]

%e convert(1002,base,3);

%e [0, 1, 0, 1, 0, 1, 1]

%e 988496761674 = 1110011000100110111111111001001101001010 (base 2)

%e 988496761674 = ..............10111111111001001101001010 (base 3)

%e 26688254956950 = 110000100010111010111111101001101010110010110 (base 2)

%e 26688254956950 = ................10111111101001101010110010110 (base 3)

%p f:=proc(n) local t2,t3,i,t0; t0:=1; t2:=convert(n,base,2); t3:=convert(n,base,3);

%p for i from 1 to nops(t3) do if t2[i] <> t3[i] then t0:=-1; fi; od; t0; end;

%p t4:=[]; for n from 1 to 20000 do if f(n) = 1 then t4:=[op(t4), n]; fi; od: t4;

%Y Cf. A175514, A178679, A178680.

%K base,nonn

%O 1,3

%A _N. J. A. Sloane_, May 30 2010

%E a(5)-a(6) from _Zak Seidov_ and _D. S. McNeil_, May 31 2010

%E a(7)-a(8) from _Alois P. Heinz_, May 31 2010

%E a(9)-a(12) from _Ray Chandler_, Jun 01 2010

%E a(11)-a(30) from _Max Alekseyev_, Jun 01 2010, Sep 21 2016, Aug 26 2020