cp's OEIS Frontend

A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.

%I A325362 #7 Feb 10 2022 13:44:59
%S A325362 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,
%T A325362 42,43,46,47,51,53,57,58,59,61,62,65,66,67,69,71,73,74,78,79,82,83,85,
%U A325362 86,87,89,93,94,95,97,101,102,103,106,107,109,110,111,113
%N A325362 Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.
%C A325362 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325362 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).
%C A325362 The enumeration of these partitions by sum is given by A007294.
%C A325362 This sequence and A025487, considered as sets, are related by the partition conjugation function A122111(.), which maps the members of either set 1:1 onto the other set. - _Peter Munn_, Feb 10 2022
%H A325362 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325362 The sequence of terms together with their prime indices begins:
%e A325362     1: {}
%e A325362     2: {1}
%e A325362     3: {2}
%e A325362     5: {3}
%e A325362     6: {1,2}
%e A325362     7: {4}
%e A325362    10: {1,3}
%e A325362    11: {5}
%e A325362    13: {6}
%e A325362    14: {1,4}
%e A325362    17: {7}
%e A325362    19: {8}
%e A325362    21: {2,4}
%e A325362    22: {1,5}
%e A325362    23: {9}
%e A325362    26: {1,6}
%e A325362    29: {10}
%e A325362    30: {1,2,3}
%e A325362    31: {11}
%e A325362    33: {2,5}
%t A325362 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325362 Select[Range[100],OrderedQ[Differences[Append[primeptn[#],0]]]&]
%Y A325362 Cf. A007294, A056239, A112798, A240026, A320348, A325327, A325360, A325364, A325367, A325390, A325394, A325400.
%Y A325362 Related to A025487 via A122111.
%K A325362 nonn
%O A325362 1,2
%A A325362 _Gus Wiseman_, May 02 2019