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A366926 Odd numbers m that can be written j * k, j >= k > 1, with floor(log_2(m)) = floor(log_2(j)) + floor(log_2(k)). 1
15, 25, 27, 45, 51, 55, 57, 63, 81, 85, 95, 99, 105, 111, 115, 117, 119, 121, 123, 125, 153, 165, 171, 175, 185, 187, 189, 195, 201, 205, 207, 209, 213, 215, 219, 221, 225, 231, 235, 237, 243, 245, 247, 249, 253, 255, 289, 297, 315, 323, 325, 333, 335, 345, 351 (list; graph; refs; listen; history; text; internal format)
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, 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
Cf. A000523.
Sequence in context: A106613 A192542 A325571 * A102802 A050692 A050693
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified April 22 09:25 EDT 2024. Contains 371894 sequences. (Running on oeis4.)