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A085078 The largest number with the prime signature of n! using primes <= n.

%I A085078 #19 Dec 30 2020 04:10:40
%S A085078 1,2,6,54,750,11250,360150,123531450,3088286250,64854011250,
%T A085078 77201350992150,65389544290351050,32637304517036749530,
%U A085078 2121424793607388719450,163349709107768931397650
%N A085078 The largest number with the prime signature of n! using primes <= n.
%C A085078 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.
%C A085078 From _Reikku Kulon_, Sep 18 2008: (Start)
%C A085078 This is n! with prime exponents reversed. Perhaps it should be denoted with an inverted exclamation mark: (inverted-!)n
%C A085078 7! = 5040 = 2^4 * 3^2 * 5^1 * 7^1
%C A085078 (inverted-!)7 = 360150 = 2^1 * 3^1 * 5^2 * 7^4 (End)
%H A085078 Amiram Eldar, <a href="/A085078/b085078.txt">Table of n, a(n) for n = 1..181</a>
%H A085078 <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>
%F A085078 a(n) = A069799(A000142(n)). - _Amiram Eldar_, Dec 30 2020
%e A085078 For n=6, 6!= 720 = 2^4*3^2*5, hence a(6) = 5^4*3^2*2 = 11250.
%e A085078 For n=8, 8! = 40320 = 2^7*3^2*5*7, hence a(8) = 7^7*5^2*3*2 = 123531450.
%t A085078 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 *)
%o A085078 (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
%Y A085078 Cf. A000142, A069799.
%Y A085078 Cf. A000040 [From _Reikku Kulon_, Sep 18 2008]
%K A085078 nonn
%O A085078 1,2
%A A085078 _Amarnath Murthy_, Jul 01 2003
%E A085078 More terms from _David Wasserman_, Jan 14 2005