A325396 Heinz numbers of integer partitions whose augmented differences are strictly decreasing.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 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, 111, 113, 114, 115

Offset

1,2

Comments

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

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The enumeration of these partitions by sum is given by A325358.

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}
   31: {11}
   33: {2,5}
   34: {1,7}

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Select[Range[100], Greater@@aug[primeptn[#]]&]

Crossrefs

A subsequence of A005117.

Cf. A056239, A093641, A112798, A320466, A325351, A325358, A325366, A325389, A325393, A325394, A325395, A325457.

Sequence in context: A333866 A325362 A339563 * A326533 A144147 A068422

Adjacent sequences: A325393 A325394 A325395 * A325397 A325398 A325399

Keyword

nonn

Author

Gus Wiseman, May 02 2019

Status

approved