A325364 Heinz numbers of integer partitions whose differences (with the last part taken to be zero) are weakly decreasing.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 35, 37, 41, 43, 47, 49, 53, 54, 55, 59, 61, 64, 65, 67, 71, 73, 75, 77, 79, 81, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 121, 125, 127, 128, 131, 133, 137, 139

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 A320509.

Links

Table of n, a(n) for n=1..63.

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

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], GreaterEqual@@Differences[Append[primeptn[#], 0]]&]

Crossrefs

Cf. A056239, A112798, A320348, A320466, A320509, A325327, A325361, A325364, A325367, A325389, A325390, A325397.

Sequence in context: A050741 A285710 A305669 * A133810 A176615 A291453

Adjacent sequences: A325361 A325362 A325363 * A325365 A325366 A325367

Keyword

nonn

Author

Gus Wiseman, May 02 2019

Status

approved