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A325370 Numbers whose prime signature has multiplicities covering an initial interval of positive integers. 7
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

First differs from A319161 in lacking 420.

The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.

Numbers whose prime signature covers an initial interval are given by A317090.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so the are Heinz numbers of integer partitions whose multiplicities have multiplicities covering an initial interval of positive integers. The enumeration of these partitions by sum is given by A325330.

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

The sequence of terms together with their prime indices begins:

    1: {}

    2: {1}

    3: {2}

    4: {1,1}

    5: {3}

    7: {4}

    8: {1,1,1}

    9: {2,2}

   11: {5}

   12: {1,1,2}

   13: {6}

   16: {1,1,1,1}

   17: {7}

   18: {1,2,2}

   19: {8}

   20: {1,1,3}

   23: {9}

   24: {1,1,1,2}

   25: {3,3}

   27: {2,2,2}

For example, the prime indices of 1890 are {1,2,2,2,3,4}, whose multiplicities give the prime signature {1,1,1,3}, and since this does not cover an initial interval (2 is missing), 1890 is not in the sequence.

MATHEMATICA

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];

Select[Range[100], normQ[Length/@Split[Sort[Last/@FactorInteger[#]]]]&]

CROSSREFS

Cf. A000009, A055932, A056239, A112798, A118914, A317081, A317089, A317090, A319161, A325326, A325330, A325337, A325369, A325371.

Sequence in context: A246716 A212165 A319161 * A329139 A130091 A119848

Adjacent sequences:  A325367 A325368 A325369 * A325371 A325372 A325373

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 02 2019

STATUS

approved

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Last modified January 20 05:22 EST 2020. Contains 331067 sequences. (Running on oeis4.)