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A325360 Heinz numbers of integer partitions whose differences are weakly increasing.

%I A325360 #9 Apr 15 2025 17:23:27
%S A325360 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,19,20,21,22,23,24,25,26,27,
%T A325360 28,29,30,31,32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,49,51,52,
%U A325360 53,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,71,73
%N A325360 Heinz numbers of integer partitions whose differences are weakly increasing.
%C A325360 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325360 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) are (-3,-2).
%C A325360 The enumeration of these partitions by sum is given by A240026.
%H A325360 Robert Price, <a href="/A325360/b325360.txt">Table of n, a(n) for n = 1..10000</a>
%H A325360 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325360 Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
%e A325360    18: {1,2,2}
%e A325360    36: {1,1,2,2}
%e A325360    50: {1,3,3}
%e A325360    54: {1,2,2,2}
%e A325360    70: {1,3,4}
%e A325360    72: {1,1,1,2,2}
%e A325360    75: {2,3,3}
%e A325360    90: {1,2,2,3}
%e A325360    98: {1,4,4}
%e A325360   100: {1,1,3,3}
%t A325360 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325360 Select[Range[100],OrderedQ[Differences[primeptn[#]]]&]
%Y A325360 Cf. A007294, A056239, A112798, A240026, A325328, A325352, A325354, A325360, A325361, A325368, A325394, A325400.
%K A325360 nonn
%O A325360 1,2
%A A325360 _Gus Wiseman_, May 02 2019