A367085 3-valuation r(n) of the terms A367083(A367084(n)+1) = 3^r(n) which are the odd terms preceded by another odd term.

1, 5, 10, 15, 20, 25, 29, 34, 39, 44, 49, 54, 58, 63, 68, 73, 78, 82, 87, 92, 97, 102, 107, 111, 116, 121, 126, 131, 135, 140, 145, 150, 155, 160, 164, 169, 174, 179, 184, 188, 193, 198, 203, 208, 213, 217, 222, 227, 232, 237, 241, 246, 251, 256, 261, 266, 270, 275, 280, 285, 290, 294, 299

Offset

0,2

Comments

These terms, A367083(A367084(n)+1) = 3^r(n), are also those which start the (maximal) groups of terms of alternating parity, (3^r(n), 4^s(n), ..., 3^(r(n+1)-1) = A367083(A367084(n+1))).

The first differences, D = (4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 4, ...) are directly related to those of A367084, viz, D(n) = (A367084(n+1)-A367084(n)+1)/2. The run lengths of the '5's are (4, 5, 4, 5, ...) with two consecutive '5' every 24 +- 1 terms.

Links

Table of n, a(n) for n=0..62.

Example

The first group (3^r, 4^s, ..., 3^r') in A367083 starts with A367083(1) = 3 = 3^1 (following the odd term A367083(A367084(0)) = 3^0 = 1), therefore a(0) = 1.
The second such group starts with A367083(8) = 3^5 (following the odd term A367083(A367084(1) = 7) = 3^4), therefore a(1) = 5.

Prog

(PARI) A367085(n)=valuation(A367083(A367084(n)+1), 3) \\ or Axxx[.+1] if vectors are used instead of the 0-indexed functions/sequences.
(PARI) /* more efficiently: */
A367085_upto(N)={my(r=1, s=1, L3=log(3), L4=log(4), A=List(r)); until(r>=N, listput(A, r += 1-s+s+=((r+4)*L3 > (s+3)*L4)+3)); Vec(A)}
(Python)
from itertools import islice
def A367085_gen(): # generator of terms
    a, b, c = 1, 4, 0
    while True:
        while (a:=a*3)<b:
            yield (c:=c+1)
        b <<= 2
        c += 1
A367085_list = list(islice(A367085_gen(), 30)) # Chai Wah Wu, Nov 09 2023

Crossrefs

Cf. A367083, A367084.

Cf. A000244 (powers of 3), A000302 (powers of 4).

Sequence in context: A007845 A299985 A313729 * A313730 A313731 A313732

Adjacent sequences: A367082 A367083 A367084 * A367086 A367087 A367088

Keyword

nonn

Author

M. F. Hasler, Nov 03 2023

Status

approved