A367083 List of powers of 3 and powers of 4, in increasing order, starting with a(0) = 3^0 = 4^0 = 1.

1, 3, 4, 9, 16, 27, 64, 81, 243, 256, 729, 1024, 2187, 4096, 6561, 16384, 19683, 59049, 65536, 177147, 262144, 531441, 1048576, 1594323, 4194304, 4782969, 14348907, 16777216, 43046721, 67108864, 129140163, 268435456, 387420489, 1073741824, 1162261467, 3486784401, 4294967296, 10460353203, 17179869184

Offset: 0

Author

M. F. Hasler, Nov 03 2023

Keywords

• nonn
• easy

Comments

The sequence mostly alternates between powers of 3 (odd terms) and powers of 4 (even terms), but after either 3 or 4 powers of 4, separated by powers of 3, there occur two consecutive powers of 3 in a row.

Sequence A367084 lists the indices n of odd terms immediately followed by another odd term. We can split the sequence of terms > 1 in groups of 7 or 9 consecutive terms (a(A367084(n)+1 .. A367084(n+1)) such that each group starts and ends with an odd term. The sequence of the group lengths will be 7, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 7, etc. There will always be 4 or 5 consecutive 9's separated by a single 7. The run lengths of the 9's are (4, 5, 4, 5, 4, 5, 4, ...) but this pattern is also slightly irregular, with two consecutive 5's occurring after every 24 (very rarely fewer) elements.

We think these patterns are important for the study of Erdős's conjecture of a positive density of Sum(Pow({3,4})) mentioned in A327621.

Crossrefs

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

Programs

Mathematica

With[{max=2*10^10},Union[3^Range[0,Log[3,max]],4^Range[0,Log[4,max]]]] (* Paul F. Marrero Romero, Nov 14 2023 *)

PARI

upto(N)={my(p(b)=[b^k|k<-[0..logint(N,b)]]);setunion(p(3), p(4))}

Python

from itertools import islice
def A367083_gen(): # generator of terms
    yield 1
    a, b = 1, 4
    while True:
        while (a:=a*3)A367083_list = list(islice(A367083_gen(),30)) # Chai Wah Wu, Nov 08 2023

Formula

Union of A000244 and A000302.