A325361 - OEIS

%I #4 May 02 2019 08:55:13

%S 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,21,22,23,25,26,27,29,30,

%T 31,32,33,34,35,37,38,39,41,43,46,47,49,50,51,53,54,55,57,58,59,61,62,

%U 64,65,67,69,70,71,73,74,75,77,79,81,82,83,85,86,87,89

%N Heinz numbers of integer partitions whose differences are weakly decreasing.

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

%C The differences of a sequence are defined as if the sequence were increasing, 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) are (-3,-2).

%C The enumeration of these partitions by sum is given by A320466.

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:

%e    12: {1,1,2}

%e    20: {1,1,3}

%e    24: {1,1,1,2}

%e    28: {1,1,4}

%e    36: {1,1,2,2}

%e    40: {1,1,1,3}

%e    42: {1,2,4}

%e    44: {1,1,5}

%e    45: {2,2,3}

%e    48: {1,1,1,1,2}

%e    52: {1,1,6}

%e    56: {1,1,1,4}

%e    60: {1,1,2,3}

%e    63: {2,2,4}

%e    66: {1,2,5}

%e    68: {1,1,7}

%e    72: {1,1,1,2,2}

%e    76: {1,1,8}

%e    78: {1,2,6}

%e    80: {1,1,1,1,3}

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

%t Select[Range[100],GreaterEqual@@Differences[primeptn[#]]&]

%Y Cf. A056239, A112798, A320466, A320509, A325328, A325352, A325456, A325457, A325360, A325361, A325364, A320466, A325368, A325389.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 02 2019