A334765 -id:A334765 - cp's OEIS frontend

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

Clark Kimberling

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

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, Compact integers and factorials, Acta Arith. 126 (2007), no. 3, 195-236.
Vladimir Shevelev, Binomial predictors, arXiv:0907.3302 [math.NT], 2009.
Lukas Spiegelhofer, Collisions of the binary and ternary sum-of-digits functions, arXiv:2105.11173 [math.NT], 2021. Proves that sequence is infinite.

Cf. A000120, A053735, A180017.

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

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

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

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

Zero prepended by Zak Seidov, May 31 2010

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions