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A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing. 12
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 78, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110, 111, 113 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).

The enumeration of these partitions by sum is given by A007294.

LINKS

Table of n, a(n) for n=1..64.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

EXAMPLE

The sequence of terms together with their prime indices begins:

    1: {}

    2: {1}

    3: {2}

    5: {3}

    6: {1,2}

    7: {4}

   10: {1,3}

   11: {5}

   13: {6}

   14: {1,4}

   17: {7}

   19: {8}

   21: {2,4}

   22: {1,5}

   23: {9}

   26: {1,6}

   29: {10}

   30: {1,2,3}

   31: {11}

   33: {2,5}

MATHEMATICA

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];

Select[Range[100], OrderedQ[Differences[Append[primeptn[#], 0]]]&]

CROSSREFS

Cf. A007294, A056239, A112798, A240026, A320348, A325327, A325360, A325364, A325367, A325390, A325394, A325400.

Sequence in context: A087006 A235991 A327906 * A325396 A326533 A144147

Adjacent sequences:  A325359 A325360 A325361 * A325363 A325364 A325365

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 02 2019

STATUS

approved

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Last modified December 5 20:45 EST 2019. Contains 329779 sequences. (Running on oeis4.)