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The phrase "a consecutive search for" is correct and usable in written English.
It can be used when describing a process of searching for something in a sequential or continuous manner.
Example: "The researchers conducted a consecutive search for evidence to support their hypothesis over several months."
Alternatives: "an ongoing search for" or "a continuous search for".
Exact(1)
With detailed analysis, proper data structures are designed so that the central computational task is formulated as a consecutive search for the minimum in the collision time space of O N2) entries, with multiple updates on O(N) entries in the same space per collision step.
Similar(59)
Top hits were used for a consecutive BLAST search.
Big sites like AllMusic.com and AllMovies.com, who are already using BiggerBoat, will see users kept on site for several consecutive searches and clicks around results.
We measured the time elapsed between consecutive searches and investigated the returned citation sets for the initial and subsequent queries, as we discuss below.
It is the most pivotal question of all as the Giants begin their seventh consecutive season searching for a replacement for Phil Simms.
A search for six consecutive nonadenine nucleotides containing at least three uridines in the intron sequence revealed that 12332 introns (86.2% of total 14305) have polypyrimidine tracts.
A search for a successor has started.
Start with a catalog search for Jamaica.
Each has a desperate search for home".
Do a web search for yourself.
Perform a web search for your question.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com