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Let    P  +   (  n  )   denote the largest prime factor of the integer   n   andP  y  +    (  n  )denote the largest prime factor   p   of   n   which satisfies    p  ⩽  y   . In this paper, first we show that the triple consecutive integers with the two patternsP  +    (  n  −  1  )   >   P  +    (  n  )   <   P  +    (  n  +  1  )andP  +    (  n  −  1  )   <   P  +    (  n  )   >   P  +    (  n  +  1  )have a positive proportion respectively. More generally, with the same methods we can prove that for any   J  ∊  Z   ,    J  ⩾  3   , the   J  ‐tuple consecutive integers with the two patternsP  +    (  n  +   j  0   )   =   min   0  ⩽  j  ⩽  J  −  1P  +    (  n  +  j  )andP  +    (  n  +   j  0   )   =   max   0  ⩽  j  ⩽  J  −  1P  +    (  n  +  j  )also have a positive proportion, respectively. Second, for    y  =   x  θwith    0  <  θ  ⩽  1    we show that there exists a positive proportion of integers   n   such thatP  y  +    (  n  )   <   P  y  +    (  n  +  1  )    . Specifically, we can prove that the proportion of integers   n   such thatP  +    (  n  )   <   P  +    (  n  +  1  )is larger than 0.1356, which improves the previous result “0.1063” of the author.

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