A297405 Binary "cubes"; numbers whose binary representation consists of three consecutive identical blocks.

7, 42, 63, 292, 365, 438, 511, 2184, 2457, 2730, 3003, 3276, 3549, 3822, 4095, 16912, 17969, 19026, 20083, 21140, 22197, 23254, 24311, 25368, 26425, 27482, 28539, 29596, 30653, 31710, 32767, 133152, 137313, 141474, 145635, 149796, 153957, 158118, 162279, 166440, 170601, 174762, 178923, 183084, 187245

Offset: 1

Author

Jeffrey Shallit, Dec 29 2017

Keywords

• nonn
• base

Comments

Alternatively, numbers of the form k*(4^n + 2^n + 1), where 2^(n-1) <= k < 2^n.

Examples

			42 in base 2 is 101010, which consists of three copies of the block "10".
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's Theorem for Binary Powers, arXiv:1801.04483 [math.NT], 2018.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A020330, which is the corresponding sequence for squares.

Subsequence of A121016.

Programs

Maple

a:= n-> (p-> n*(1+2^p+4^p))(1+ilog2(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 29 2017

Mathematica

bc[n_]:=FromDigits[Join[n,n,n],2]; Flatten[Table[bc/@Select[Tuples[ {1,0},n],#[[1]] == 1&],{n,6}]]//Union (* Harvey P. Dale, Oct 09 2021 *)

PARI

a(n) = n=binary(n); fromdigits(concat([n, n, n]) , 2) \\ Iain Fox, Jul 04 2022

Python

def a(n): return int(bin(n)[2:]*3, 2)
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Jul 04 2022
# Alternative:
def A297405(n):
    p = n.bit_length()
    return n * (1 + 2**p + 4**p)
print([A297405(n) for n in range(1, 46)])  # Peter Luschny, Jul 05 2022

Formula

a(n) = n*(1 + 2^p + 4^p) with p = 1 + floor(log_2(n)). - Alois P. Heinz, Dec 29 2017

G.f.: (7*x + Sum_{n>=1} (4^n + 3*8^n + (2^n + 2*4^n - 3*8^n)*x)*x^(2^n))/(1-x)^2. - Robert Israel, Dec 31 2017