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exibindo a semana conforme a data


Em vez de usar o WEEKDAY, use WEEKOFYEAR 
SELECT WEEKOFYEAR(date) AS week, date
FROM myTable
WHERE date BETWEEN start_date AND end_date
ORDER BY week

ATUALIZAÇÃO Esta pergunta é a fonte 
SELECT  (WEEK(date, 5) -
        WEEK(DATE_SUB(date, INTERVAL DAYOFMONTH(date) - 1 DAY), 5) + 1) as week, date
    FROM myTable
    WHERE date BETWEEN start_date AND end_date
    ORDER BY week

ATUALIZAÇÃO 2 
SELECT  (WEEK(date, 1) -
        WEEK(DATE_SUB(date, INTERVAL DAYOFMONTH(date) - 1 DAY), 1) + 1) as week, date
    FROM myTable
    WHERE date BETWEEN start_date AND end_date
    ORDER BY week