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

1,2

Comments

The parts of each partition are listed in increasing order.

Links

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A327392 A112798 A187846 * A029288 A238899 A187207

Adjacent sequences: A181084 A181085 A181086 * A181088 A181089 A181090

Keyword

nonn,look

Author

Alois P. Heinz, Jan 23 2011

Status

approved