A181087 Partitions of n in the order of increasing smallest numbers of prime signatures.

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, 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, 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

Offset: 1

Author

Alois P. Heinz, Jan 23 2011

Keywords

• nonn
• look

Comments

The parts of each partition are listed in increasing order.

Examples

			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.
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], ...
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
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..18132

Crossrefs

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

Programs

Mathematica

DeleteDuplicates[Map[Sort[Map[Last, FactorInteger[#]]] &, Range[1000]]] // Grid (* Geoffrey Critzer, Nov 27 2015 *)

Sage

def A181087_build(w):
    seen = set()
    a = []
    for n in PositiveIntegers():
        psig = tuple(sorted(m for p,m in factor(n)))
        if psig not in seen:
            a.extend(psig)
            seen.add(psig)
            if len(a) >= w: return a  # D. S. McNeil, Jan 23 2011