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%I #14 Jul 15 2018 13:39:57
%S 1,6,30,36,210,180,216,2310,1260,900,1080,1296,30030,13860,6300,7560,
%T 5400,6480,7776,510510,180180,69300,83160,44100,37800,45360,27000,
%U 32400,38880,46656,9699690,3063060,900900,1081080,485100,415800,498960,264600,189000
%N Leading least prime signatures, ordered by forming the product of primorials greater than 2 with multiplicities given by the canonical sequence of partitions.
%C From _Jack W Grahl_, Jul 06 2018: (Start)
%C The least prime signatures (A025487) are the smallest numbers with a given 'prime signature'. For example, 24 = 2^3 * 3 is the smallest number consisting of the cube of a prime multiplied by a prime. They can be expressed as products 2^(k1) * 3^(k2) * 5(k3) * ..., where k1 >= k2 >= k3 >= ...
%C These can also be defined as all products of primorials. Here the primorials (A002110) are the products of the first n primes. So 24 = 2 * 2 * 6.
%C The leading least prime signatures (A056153) are the least prime signatures k such that k/2 is not a least prime signature. They can be expressed as products 2^(k1) * 3^(k2) * 5(k3) * ..., where k1 = k2 >= k3 >= ... (note the first operator is equality). A056153 lists these in increasing order.
%C They can also be defined as all products of primorials 6 or greater. This sequence lists the leading least prime signatures in an ordering derived from this definition. The canonical sequence of partitions maps to this sequence under a mapping which sends 1 -> 6, 2 -> 30, 3 -> 210, etc., and then forms the product of these terms. Thus the first few partitions are [], [1], [2], [1,1], [3], [2,1] and so the first terms of this sequence are 1, 6, 30, 6 * 6 = 36, 210, 30 * 6 = 180.
%C The previous description described this sequence as the 'leading least prime signatures ordered as in A063008'. This was in error. A063008 gives a different ordering of A025487, also based on the canonical sequence of partitions, but the definition is different from this sequence and the terms do not appear in the same order (with the transposition of 216 and 2310 being the first discrepancy). (End)
%H Jack W Grahl, <a href="/A062515/b062515.txt">Table of n, a(n) for n = 0..1000</a>
%e Values in A025487 can be generated via powers of two as follows:
%e 1
%e 2
%e 4,6
%e 8,12
%e 16,24,30
%e 32,48,60,36
%e 64,96,120,72
%e 128,192,240,144,210,180,216
%e a(3) = 36 because we can write [1,1] and associate this exponent vector with 6*6
%o (Haskell)
%o import Data.List(inits)
%o primes :: [Integer]
%o primes = 2 : 3 : filter (\a -> all (not . divides a) (takeWhile (\x -> x <= a `div` 2) primes)) [4..]
%o where
%o divides a b = a `mod` b == 0
%o primorials :: [Integer]
%o primorials = map product $ inits primes
%o partitions :: [[Integer]]
%o partitions = concat $ map (partitions_of_n) [0..]
%o partitions_of_n :: Integer -> [[Integer]]
%o partitions_of_n n = partitions_at_most n n
%o partitions_at_most :: Integer -> Integer -> [[Integer]]
%o partitions_at_most _ 0 = [[]]
%o partitions_at_most 0 _ = []
%o partitions_at_most m n = concat $ map (\k -> map ([k] ++) (partitions_at_most k (n-k))) ( reverse [1..(min m n)])
%o a062515 :: [Integer]
%o a062515 = map primorial_signature partitions
%o where
%o primorial_signature p = product $ map ((drop 1 primorials) !!) (map fromIntegral p)
%o -- _Jack W Grahl_, Jul 06 2018
%Y Cf. A002110, A025487, A056153 and A063008.
%K easy,nice,nonn
%O 0,2
%A _Alford Arnold_, Jul 10 2001
%E Clarified and extended by _Jack W Grahl_, Jul 06 2018
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