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A181087 Partitions of n in the order of increasing smallest numbers of prime signatures. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A131375 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

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Last modified December 10 15:13 EST 2016. Contains 279003 sequences.