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A336414
Number of divisors of n! with distinct prime multiplicities.
24
1, 1, 2, 3, 7, 10, 20, 27, 48, 86, 147, 195, 311, 390, 595, 1031, 1459, 1791, 2637, 3134, 4747, 7312, 10766, 12633, 16785, 26377, 36142, 48931, 71144, 82591, 112308, 128023, 155523, 231049, 304326, 459203, 568095, 642446, 812245, 1137063, 1441067, 1612998, 2193307, 2429362
OFFSET
0,3
COMMENTS
A number has distinct prime multiplicities iff its prime signature is strict.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..6245 (n = 0..94 from David A. Corneth)
FORMULA
a(n) = A181796(n!).
EXAMPLE
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.
1: () 20: (2,1) | 6: (1,1)
2: (1) 24: (3,1) | 10: (1,1)
3: (1) 40: (3,1) | 15: (1,1)
4: (2) 45: (2,1) | 30: (1,1,1)
5: (1) 48: (4,1) | 36: (2,2)
8: (3) 72: (3,2) | 60: (2,1,1)
9: (2) 80: (4,1) | 90: (1,2,1)
12: (2,1) 144: (4,2) | 120: (3,1,1)
16: (4) 360: (3,2,1) | 180: (2,2,1)
18: (1,2) 720: (4,2,1) | 240: (4,1,1)
MATHEMATICA
Table[Length[Select[Divisors[n!], UnsameQ@@Last/@FactorInteger[#]&]], {n, 0, 15}]
PROG
(PARI) a(n) = sumdiv(n!, d, my(ex=factor(d)[, 2]); #vecsort(ex, , 8) == #ex); \\ Michel Marcus, Jul 24 2020
CROSSREFS
Perfect-powers are A001597, with complement A007916.
Numbers with distinct prime multiplicities are A130091.
Divisors with distinct prime multiplicities are counted by A181796.
The maximum divisor with distinct prime multiplicities is A327498.
Divisors of n! with equal prime multiplicities are counted by A336415.
Sequence in context: A173132 A351704 A320675 * A095010 A295723 A306008
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 22 2020
EXTENSIONS
a(21)-a(41) from Alois P. Heinz, Jul 24 2020
STATUS
approved