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como calcular o número total de sábado e domingo entre duas datas no mysql


Se você não tiver uma tabela com todas as datas entre a data de início e a data de término (inclusive), precisará adotar uma consulta que criará primeiro todas as datas entre o intervalo de datas fornecido (inclusive). Em seguida, use WEEKDAY função de MySQL para verificar se o dia é Saturday ou Sunday
SELECT 
dateTable.Day,
DAYNAME(dateTable.Day) AS dayName
FROM 
(   SELECT ADDDATE('2016-07-01', INTERVAL @i:[email protected]+1 DAY) AS DAY
    FROM (
        SELECT a.a
        FROM (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS a
        CROSS JOIN (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS b
        CROSS JOIN (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS c
    ) a
    JOIN (SELECT @i := -1) r1
    WHERE 
    @i < DATEDIFF('2016-08-01', '2016-07-01')

) AS dateTable
WHERE WEEKDAY(dateTable.Day) IN (5,6)
ORDER BY dateTable.Day;

DEMONSTRAÇÃO DE TRABALHO

Observação: WEEKDAY retorna o índice do dia da semana para a data (0 =segunda-feira, 1 =terça-feira, … 6 =domingo).

EDITAR:

Se você precisar apenas de count :
SELECT 
COUNT(*) AS total
FROM 
(   SELECT ADDDATE('2016-07-01', INTERVAL @i:[email protected]+1 DAY) AS DAY
    FROM (
        SELECT a.a
        FROM (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS a
        CROSS JOIN (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS b
        CROSS JOIN (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS c
    ) a
    JOIN (SELECT @i := -1) r1
    WHERE 
    @i < DATEDIFF('2016-08-01', '2016-07-01')

) AS dateTable
WHERE WEEKDAY(dateTable.Day) IN (5,6)

Demonstração