OFFSET
1,1
COMMENTS
These are odd numbers that factorize nontrivially as j * k, such that the associated multiplication operation in binary generates no carry in the most significant position.
We exclude even numbers from consideration here as every even m = 2 * k would satisfy the equation given [since log_2(2) = 1.0, so log_2(m) = 1.0 + log_2(k), so floor(log_2(m)) = 1 + floor(log_2(k)) and 1 = floor(log_2(2))].
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20, labeling the x-axis with indices of the first term m that exceeds 2^i, and that m labeling the y-axis.
Michael De Vlieger, 1024 X 1024 raster showing a(n), n = 1..2^20, showing prime powers in gold, squarefree numbers in green, and numbers neither squarefree nor prime powers in blue. Exhibits the prevalence of the last mentioned numbers early in the interval (2^i..2^(i+1)).
Michael De Vlieger, Binary expansion of a(n), n = 1..1024, shown left to right, with least significant bit at bottom, most significant at top, showing 1s in black, 0s in white.
EXAMPLE
Let f(x) = floor(log_2(x)).
9 is not in the sequence since 9 = 3 * 3, f(9, 3, 3) = {3, 1, 1}, but 1 and 1 do not sum to 3.
a(1) = 15 = 3 * 5 since f(15, 3, 5) = {3, 1, 2}, 3 = 1 + 2.
a(2) = 25 = 5 * 5 since f(25, 5, 5) = {4, 2, 2}, 4 = 2 + 2.
a(3) = 27 = 3 * 9 since f(27, 3, 9) = {4, 1, 3}, 4 = 1 + 3.
a(4) = 45 = 5 * 9 since f(45, 5, 9) = {5, 2, 3}, 5 = 2 + 3.
a(13) = 105 = 3 * 35 = 5 * 21; both these combinations satisfy the condition for entry.
Odd primes p are not in the sequence since they cannot be written j * k, j >= k > 1.
MATHEMATICA
Select[Select[Range[1, 352, 2], CompositeQ],
Function[k, AnyTrue[Total /@ Transpose@ {
Floor@ Log2@ #1[[1 ;; #2]],
Floor@ Log2@ Reverse@ #1[[-#2 ;; -1]]}, # == Floor@ Log2[k] &] & @@
{#, Ceiling[Length[#]/2]} &@ Divisors[k][[2 ;; -2]] ] ] (* Michael De Vlieger, Oct 28 2023 *)
PROG
(PARI) is(n) = if(n%2==0, return(0)); my(d=divisors(n), qb=logint(n, 2)); for(i = 2, (#d+1)\2, if(logint(d[i], 2)+logint(d[#d+1-i], 2) == qb, return(1))); 0 \\ David A. Corneth, Oct 29 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Munn and Michael De Vlieger, Oct 28 2023
STATUS
approved