

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
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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 xaxis with indices of the first term m that exceeds 2^i, and that m labeling the yaxis.
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+1i], 2) == qb, return(1))); 0 \\ David A. Corneth, Oct 29 2023


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



