

A037301


Numbers k such that the sum of base2 digits of k equals the sum of base3 digits of k.


11



0, 1, 6, 7, 10, 11, 12, 13, 18, 19, 21, 36, 37, 46, 47, 58, 59, 60, 61, 86, 92, 102, 103, 114, 115, 120, 121, 166, 167, 172, 173, 180, 181, 198, 199, 216, 217, 222, 223, 261, 273, 282, 283, 285, 298, 299, 300, 301, 306, 307, 309, 318
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OFFSET

1,3


COMMENTS

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
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
A053735(a(n)) = A000120(a(n)); A180017(a(n)) = 0.  Reinhard Zumkeller, Aug 06 2010
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
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


LINKS

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, Compact integers and factorials, Acta Arith. 126 (2007), no. 3, 195236.
Vladimir Shevelev, Binomial predictors, arXiv:0907.3302 [math.NT], 2009.


MATHEMATICA

Select[ Range@ 320, Total@ IntegerDigits[#, 2] == Total@ IntegerDigits[#, 3] &] (* Robert G. Wilson v, Oct 24 2014 *)


PROG

(PARI) is(n)=sumdigits(n, 3)==hammingweight(n) \\ Charles R Greathouse IV, May 21 2015


CROSSREFS

Cf. A001316, A051638, A212222, A330904, A334765.
Sequence in context: A286473 A165363 A006364 * A163247 A085267 A118957
Adjacent sequences: A037298 A037299 A037300 * A037302 A037303 A037304


KEYWORD

nonn,base


AUTHOR

Clark Kimberling


EXTENSIONS

Zero prepended by Zak Seidov, May 31 2010


STATUS

approved



