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A182856 a(0) = 1; for n > 0, a(n) = smallest positive integer whose prime signature contains, for k = 1 to n, exactly one positive number appearing exactly k times.

%I A182856 #31 Jul 08 2023 20:57:55
%S A182856 1,2,60,1801800,11657093261814000,7167827541370578634694420017740000,
%T A182856 291943326350524088652207164949980988754136887856059678357800000
%N A182856 a(0) = 1; for n > 0, a(n) = smallest positive integer whose prime signature contains, for k = 1 to n, exactly one positive number appearing exactly k times.
%C A182856 a(n) = smallest number m such that A181819(m) = A006939(n).  a(n) belongs to A182855 iff n > 1.
%C A182856 Next term has 105 digits.
%C A182856 Smallest number k with A323022(k) = n, where A323022(m) is the number of distinct multiplicities in the prime signature of m. - _Gus Wiseman_, Jan 03 2019
%H A182856 David A. Corneth, <a href="/A182856/b182856.txt">Table of n, a(n) for n = 0..12</a>
%H A182856 Dario Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>
%F A182856 Partial products of A113511.
%F A182856 log a(n) ~ (1/3) n^3 log n. [_Charles R Greathouse IV_, Jan 13 2012]
%F A182856 A001222(a(n)) = A000292(n). - _Gus Wiseman_, Jan 03 2019
%F A182856 a(0) = 1; a(n + 1) = A002110(binomial(n + 2, 2)) * a(n). - _David A. Corneth_, Jan 03 2019
%e A182856 The canonical prime factorization of a(3) = 1801800 is 2^3*3^2*5^2*7*11*13. The prime signature of 1801800 is therefore (3,2,2,1,1,1). Note that (3,2,2,1,1,1) contains exactly one number that appears once (3), one number that appears twice (2), and one number that appears three times (1).
%t A182856 Table[Product[Times@@Prime[i*(i-1)/2+Ceiling[Range[i*(n-i)]/(n-i)]],{i,n-1}],{n,6}] (* _Gus Wiseman_, Jan 03 2019 *)
%o A182856 (PARI) a(n) = if(n == 0, return(1)); my(f = matrix(binomial(n+1,2), 2)); f[, 1] = primes(#f~ )~; f[, 2] = Vecrev(concat(vector(n, i, vector(n+1-i, j, i))))~; factorback(f) \\ _David A. Corneth_, Jan 03 2019
%Y A182856 Cf. A006939, A181819, A182855.
%Y A182856 Cf. A056239, A059404, A062770, A071625, A118914, A181821, A182857, A323022.
%K A182856 nonn,easy
%O A182856 0,2
%A A182856 _Matthew Vandermast_, Jan 05 2011