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%I #31 Aug 25 2020 21:09:19
%S 1,1,2,3,7,10,20,27,48,86,147,195,311,390,595,1031,1459,1791,2637,
%T 3134,4747,7312,10766,12633,16785,26377,36142,48931,71144,82591,
%U 112308,128023,155523,231049,304326,459203,568095,642446,812245,1137063,1441067,1612998,2193307,2429362
%N Number of divisors of n! with distinct prime multiplicities.
%C A number has distinct prime multiplicities iff its prime signature is strict.
%H Chai Wah Wu, <a href="/A336414/b336414.txt">Table of n, a(n) for n = 0..6245</a> (n = 0..94 from David A. Corneth)
%F a(n) = A181796(n!).
%e The first and second columns below are the a(6) = 20 counted divisors of 6! together with their prime signatures. The third column shows the A000005(6!) - a(6) = 10 remaining divisors.
%e 1: () 20: (2,1) | 6: (1,1)
%e 2: (1) 24: (3,1) | 10: (1,1)
%e 3: (1) 40: (3,1) | 15: (1,1)
%e 4: (2) 45: (2,1) | 30: (1,1,1)
%e 5: (1) 48: (4,1) | 36: (2,2)
%e 8: (3) 72: (3,2) | 60: (2,1,1)
%e 9: (2) 80: (4,1) | 90: (1,2,1)
%e 12: (2,1) 144: (4,2) | 120: (3,1,1)
%e 16: (4) 360: (3,2,1) | 180: (2,2,1)
%e 18: (1,2) 720: (4,2,1) | 240: (4,1,1)
%t Table[Length[Select[Divisors[n!],UnsameQ@@Last/@FactorInteger[#]&]],{n,0,15}]
%o (PARI) a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); #vecsort(ex,,8) == #ex); \\ _Michel Marcus_, Jul 24 2020
%Y Perfect-powers are A001597, with complement A007916.
%Y Numbers with distinct prime multiplicities are A130091.
%Y Divisors with distinct prime multiplicities are counted by A181796.
%Y The maximum divisor with distinct prime multiplicities is A327498.
%Y Divisors of n! with equal prime multiplicities are counted by A336415.
%Y Cf. A000005, A098859, A118914, A124010, A327527, A336424, A336500, A336568.
%Y Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jul 22 2020
%E a(21)-a(41) from _Alois P. Heinz_, Jul 24 2020