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A316532 Leading least prime signatures, ordered by the underlying partitions, as in A063008.

%I A316532 #7 Jul 15 2018 13:42:13
%S A316532 1,6,30,36,210,180,2310,216,900,1260,30030,1080,6300,13860,510510,
%T A316532 1296,5400,7560,44100,69300,180180,9699690,6480,27000,37800,83160,
%U A316532 485100,900900,3063060,223092870,7776,32400,45360,189000,264600,415800,1081080,5336100
%N A316532 Leading least prime signatures, ordered by the underlying partitions, as in A063008.
%C A316532 The sequence A063008 gives the least number with each prime signature, ordered by the underlying partition. This sequence is a subsequence which only includes those prime signatures M for which M/2 is not a prime signature, the so-called 'leading' least prime signatures.
%C A316532 This sequence is therefore constructed by taking the partitions first in increasing order of their sum, then in decreasing order of the first term, then decreasing order of the second term, etc. We drop all partitions, except the empty partition, where the first term and the second term are different. Then we map (m1, m2, m3, ..., mk) to 2^m1 * 3^m2 * ... * pk^mk to give the terms of this sequence.
%C A316532 The sequence A062515 had a description which suggested that it had been confused with this sequence. They are the same leading least prime signatures, but in a different order, given by a different construction using integer partitions.
%e A316532 The first few partitions are [], [1,1], [1,1,1], [2,2], [1,1,1,1]. So the first few terms are 1, 2 * 3 = 6, 2 * 3 * 5 = 30, 2^2 * 3^2 = 36, 2 * 3 * 5 * 7 = 210.
%o A316532 (Haskell)
%o A316532 primes :: [Integer]
%o A316532 primes = 2 : 3 : filter (\a -> all (not . divides a) (takeWhile (\x -> x <= a `div` 2) primes)) [4..]
%o A316532 divides :: Integer -> Integer -> Bool
%o A316532 divides a b = a `mod` b == 0
%o A316532 partitions :: [[Integer]]
%o A316532 partitions = concat $ map (partitions_of_n) [0..]
%o A316532 partitions_of_n :: Integer -> [[Integer]]
%o A316532 partitions_of_n n = partitions_at_most n n
%o A316532 partitions_at_most :: Integer -> Integer -> [[Integer]]
%o A316532 partitions_at_most _ 0 = [[]]
%o A316532 partitions_at_most 0 _ = []
%o A316532 partitions_at_most m n = concat $ map (\k -> map ([k] ++) (partitions_at_most k (n-k))) ( reverse [1..(min m n)])
%o A316532 prime_signature :: [Integer] -> Integer
%o A316532 prime_signature p = product $ zipWith (^) primes p
%o A316532 seq :: [Integer]
%o A316532 seq = map prime_signature $ filter compare_first_second partitions
%o A316532     where
%o A316532   compare_first_second p
%o A316532         | length p == 0 = True
%o A316532         | length p == 1 = False
%o A316532         | otherwise = p!!0 == p!!1
%Y A316532 Subsequence of A063008. A re-ordering of A062515, also of A056153. Cf A025487.
%K A316532 nonn,easy
%O A316532 0,2
%A A316532 _Jack W Grahl_, Jul 06 2018