%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