A181087 - OEIS

%I #30 Mar 08 2020 00:05:28

%S 1,2,1,1,3,1,2,4,1,3,1,1,1,5,2,2,1,4,1,1,2,6,2,3,1,5,1,1,3,7,2,4,1,2,

%T 2,1,6,1,1,1,1,3,3,1,1,4,8,2,5,1,2,3,1,7,1,1,1,2,3,4,1,1,5,9,2,6,1,2,

%U 4,1,8,1,1,1,3,3,5,2,2,2,1,1,6,10,1,3,3,2,7,1,1,2,2,4,4,1,2,5,1,9,1,1,1,4,3,6,2,2,3,1,1,7,11,1,3,4,2,8,1,1

%N Partitions of n in the order of increasing smallest numbers of prime signatures.

%C The parts of each partition are listed in increasing order.

%H Alois P. Heinz, <a href="/A181087/b181087.txt">Table of n, a(n) for n = 1..18132</a>

%e Smallest number with prime signature [1,1,1] is 2^1*3^1*5^1 = 30, the smallest number for [4] is 2^4 = 16, and thus [4] < [1,1,1] in this order.

%e First partitions in the order of increasing smallest numbers of prime signatures are: [1], [2], [1,1], [3], [1,2], [4], [1,3], [1,1,1], [5], [2,2], [1,4], [1,1,2], [6], [2,3], [1,5], [1,1,3], [7], [2,4], ...

%e Smallest numbers with these prime signatures are:  2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, ... A025487

%t DeleteDuplicates[Map[Sort[Map[Last, FactorInteger[#]]] &, Range[1000]]] // Grid (* _Geoffrey Critzer_, Nov 27 2015 *)

%o (Sage)

%o def A181087_build(w):

%o     seen = set()

%o     a = []

%o     for n in PositiveIntegers():

%o         psig = tuple(sorted(m for p,m in factor(n)))

%o         if psig not in seen:

%o             a.extend(psig)

%o             seen.add(psig)

%o             if len(a) >= w: return a  # _D. S. McNeil_, Jan 23 2011

%Y Cf. A036036, A036037, A080576, A025487, A095904.

%K nonn,look

%O 1,2

%A _Alois P. Heinz_, Jan 23 2011