A037301 Numbers whose base-2 and base-3 expansions have the same digit sum.

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

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

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

Lukas Spiegelhofer, Collisions of the binary and ternary sum-of-digits functions, arXiv:2105.11173 [math.NT], 2021.

Vladimir Shevelev, Compact integers and factorials, Acta Arith. 126 (2007), no. 3, 195-236.

Vladimir Shevelev, Binomial predictors, arXiv:0907.3302 [math.NT], 2009.

Formula

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

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. A000120, A053735, A180017.

Cf. A001316, A051638, A212222, A330904, A334765.

Sequence in context: A358432 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