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%I #77 Jun 21 2021 02:49:50
%S 0,1,6,7,10,11,12,13,18,19,21,36,37,46,47,58,59,60,61,86,92,102,103,
%T 114,115,120,121,166,167,172,173,180,181,198,199,216,217,222,223,261,
%U 273,282,283,285,298,299,300,301,306,307,309,318
%N Numbers whose base-2 and base-3 expansions have the same digit sum.
%C If Sum_{i=0..k} (binomial(k,i) mod 2) == Sum_{i=0..k} (binomial(k,i) mod 3) then k is in the sequence. (The converse does not hold.) - _Benoit Cloitre_, Nov 16 2003
%C Problem: To prove that the sequence is infinite. A generalization: Let s_m(k) denote the sum of digits of k in base m; does the Diophantine equation s_p(k) = s_q(k), where p,q are fixed distinct primes, have infinitely many solutions? - _Vladimir Shevelev_, Jul 30 2009
%C Also, numbers k such that the exponent of the largest power of 2 dividing k! is exactly twice the exponent of the largest power of 3 dividing k!. - _Ivan Neretin_, Mar 08 2015
%C a(5) = 10, a(6) = 11, a(7) = 12 and a(8) = 13 is the first time that four consecutive terms appear in this sequence. Conjecture: There is no occurrence of five or more consecutive terms of a(n). Tested by exhaustive search up to a(n) = 3^29. - _Thomas König_, Aug 15 2020
%H Stanislav Sykora, <a href="/A037301/b037301.txt">Table of n, a(n) for n = 1..10000</a>
%H Lukas Spiegelhofer, <a href="https://arxiv.org/abs/2105.11173">Collisions of the binary and ternary sum-of-digits functions</a>, arXiv:2105.11173 [math.NT], 2021.
%H Vladimir Shevelev, <a href="http://dx.doi.org/10.4064/aa126-3-1">Compact integers and factorials</a>, Acta Arith. 126 (2007), no. 3, 195-236.
%H Vladimir Shevelev, <a href="http://arXiv.org/abs/0907.3302">Binomial predictors</a>, arXiv:0907.3302 [math.NT], 2009.
%F A053735(a(n)) = A000120(a(n)); A180017(a(n)) = 0. - _Reinhard Zumkeller_, Aug 06 2010
%t Select[ Range@ 320, Total@ IntegerDigits[#, 2] == Total@ IntegerDigits[#, 3] &] (* _Robert G. Wilson v_, Oct 24 2014 *)
%o (PARI) is(n)=sumdigits(n,3)==hammingweight(n) \\ _Charles R Greathouse IV_, May 21 2015
%Y Cf. A000120, A053735, A180017.
%Y Cf. A001316, A051638, A212222, A330904, A334765.
%K nonn,base
%O 1,3
%A _Clark Kimberling_
%E Zero prepended by _Zak Seidov_, May 31 2010
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