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a(n) is the smallest k such that k!'s prime(n)-smooth part is less than its prime(n+1)-rough part.
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%I #7 Feb 03 2023 08:16:51

%S 3,21,47,111,186,293,437,619,830,1070,1358,1662,2019,2428,2903,3373,

%T 3908,4493,5113,5791,6506,7325,8150,9043,9942,10929,11983,13089,14303,

%U 15591,16845,18143,19535,21003,22488,24046,25693,27333,29119,30905,32741,34764,36734

%N a(n) is the smallest k such that k!'s prime(n)-smooth part is less than its prime(n+1)-rough part.

%e For n=1, the prime(1)-smooth part of k! (i.e., the 2-smooth part) is the part whose prime factors are 2's; the prime(2)-rough part (i.e., the 3-rough part) is the odd part. For k = 1..3, we have

%e k k! 2-smooth part 3-rough part

%e - -- ------------- ------------

%e 1 1 1 = 1

%e 2 2 2 > 1

%e 3 6 2 < 3

%e so a(1)=3.

%e Similarly, for n=2, we have

%e k 3-smooth part 5-rough part

%e -- ------------- ------------

%e 1 1 = 1

%e 2 2 > 1

%e 3 6 > 1

%e 4 24 > 1

%e 5 24 > 5

%e . . . .

%e . . . .

%e 18 429981696 > 14889875

%e 19 429981696 > 282907625

%e 20 1719926784 > 1414538125

%e 21 5159780352 < 9901766875

%e so a(2)=21.

%Y Cf. A000142.

%K nonn

%O 1,1

%A _Jon E. Schoenfield_, Feb 03 2023