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A085078
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The largest number with the prime signature of n! using primes <= n.
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1
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1, 2, 6, 54, 750, 11250, 360150, 123531450, 3088286250, 64854011250, 77201350992150, 65389544290351050, 32637304517036749530, 2121424793607388719450, 163349709107768931397650
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OFFSET
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1,2
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COMMENTS
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n! is the smallest number with that prime signature. E.g. 720 = 2^4*3^2*5. (Can we name a(n) as the eldest brother of n!?) Subsidiary sequence: Total number of distinct numbers with prime signature that of n! having prime divisors less than or equal to n.
This is n! with prime exponents reversed. Perhaps it should be denoted with an inverted exclamation mark: (inverted-!)n
7! = 5040 = 2^4 * 3^2 * 5^1 * 7^1
(inverted-!)7 = 360150 = 2^1 * 3^1 * 5^2 * 7^4 (End)
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LINKS
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FORMULA
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EXAMPLE
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For n=6, 6!= 720 = 2^4*3^2*5, hence a(6) = 5^4*3^2*2 = 11250.
For n=8, 8! = 40320 = 2^7*3^2*5*7, hence a(8) = 7^7*5^2*3*2 = 123531450.
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MATHEMATICA
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a[n_] := Module[{f = FactorInteger[n!], p, e}, p = First /@ f; e = Last /@ f; Times @@ (p^Reverse[e])]; Array[a, 15] (* Amiram Eldar, Dec 30 2020 *)
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PROG
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(PARI) for (n = 1, 20, f = factor(n!); c = matsize(f)[1]; a = prod(i = 1, c, f[i, 1]^f[c + 1 - i, 2]); print(a)); \\ David Wasserman, Jan 14 2005
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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