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A367086
Exponents k > 0 such that the interval [4^(k-1), 4^k] contains two powers of 3.
0
1, 4, 8, 12, 16, 20, 23, 27, 31, 35, 39, 43, 46, 50, 54, 58, 62, 65, 69, 73, 77, 81, 85, 88, 92, 96, 100, 104, 107, 111, 115, 119, 123, 127, 130, 134, 138, 142, 146, 149, 153, 157, 161, 165, 169, 172, 176, 180, 184, 188, 191, 195, 199, 203, 207, 211, 214, 218, 222, 226, 230, 233, 237, 241
OFFSET
0,2
COMMENTS
This is a list or set of numbers but at the same time a function of n related to other sequences A367083 - A367085 that all use the same index n starting at offset 0, which explains why this sequence also starts at offset 0.
The list of powers of 3 and powers of 4 by increasing size is A367083 = (1; 3^1, 4^1, 3^2, 4^2, 3^3, 4^3, 3^4; 3^5, 4^4, 3^6, 4^5, 3^7, 4^6, 3^8, 4^7, 3^9; 3^10, ...). That list can be split into groups (3^r, 4^s, ..., 3^r') of either 4+3 = 7 or 5+4 = 9 terms which start and end with a power of three. Otherwise said, the end of one group and the start of the next group are two consecutive powers of 3 that lie between two consecutive powers of 4.
This sequence lists the exponent of the first power of 4 in each group: these are exactly the exponents k of powers of 4 such that there are two powers of 3 in the interval [4^(k-1), 4^k].
The first differences, D = (3, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3, ...) are directly related to those of A367084 and A367085, viz, D(n) = (A367084(n+1)-A367084(n)-1)/2 = A367085(n+1)-A367085(n)-1. The run lengths of the '4's are (4, 5, 4, 5, ...) with two consecutive '5's every 24 +- 1 terms.
FORMULA
a(n) = A235127( A367083( A367084(n)+2 )), where A235127 is the 4-valuation.
a(n) = 1 + floor(n/log_3(4/3)) = 1 + floor(n/(log_3(4) - 1)).
EXAMPLE
The smallest power 4^s such that the interval [4^(s-1), 4^s] contains two powers of 3 is 4^1, i.e., s = 1, where [4^0, 4^1] contains 3^0 and 3^1. Hence a(0) = 1. (This is also the exponent of the smallest power of 4 in the first group of the form (3^r, 4^s, ..., 3^r') in A367083, namely: (3^1, 4^1, 3^2, 4^2, 3^3, 4^3, 3^4).)
The next larger power of 4 with this property is 4^4, hence a(1) = 4, where [4^3, 4^4] contains 3^4 and 3^5. This is also the least exponent of a power of 4 in the second group (3^5, 4^4, 3^6, 4^5, ..., 3^9), which is marked on the left in the table below.
.
Numbers of the forms
3^r 4^s
------ ------
...
| 16
| 27 __________ the interval
| 64 | [4^3, 4^4]
\____ 81 | includes two
/ 243 | powers of 3,
2 | ____ 256 _| so 4 is a term
n | 729 of this sequence
d | 1024
| 2187
g | 4096
r | 6561 __________ the interval
p | 16384 | [4^7, 4^8]
\_ 19683 | includes two
/ 59049 | powers of 3,
| __ 65536 _| so 8 is a term
| 177147 of this sequence
| 262144
| 531441
...
PROG
(PARI) A367086_upto(N)={my(r=1, s=1, L3=log(3), L4=log(4), A=List(s)); until(r>=N, listput(A, s-=1+r-r+=((r+4)*L3 > (s+3)*L4)+4)); Vec(A)}
(Python)
from itertools import islice
def A367086_gen(): # generator of terms
a, b, c, i = 1, 4, -1, 1
while True:
while (a:=a*3)<b:
c += 1
yield i
b <<= 2
i += 1
c += 2
A367086_list = list(islice(A367086_gen(), 30)) # Chai Wah Wu, Nov 18 2023
CROSSREFS
Cf. A000244 (powers of 3), A000302 (powers of 4).
Sequence in context: A311122 A190714 A311123 * A285466 A274141 A086133
KEYWORD
nonn
AUTHOR
M. F. Hasler, Nov 03 2023
STATUS
approved