cp's OEIS Frontend

A322529 Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

%I A322529 #16 Aug 17 2024 03:20:47
%S A322529 1,1,2,2,1,3,2,3,2,2,4,2,3,3,4,4,4,3,5,4,5,6,6,6,6,6,8,6,7,9,8,11,8,
%T A322529 11,11,11,12,13,13,15,13,17,17,18,18,17,20,22,21,24,24,24,26,29,28,33,
%U A322529 30,35,34,38,38,45,42,43,45,48,52,54,55,59,59,65,65,72,73
%N A322529 Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.
%C A322529 Such a partition must be strict (unless it is all 1's) and its parts must also be squarefree.
%H A322529 Lucas A. Brown, <a href="/A322529/b322529.txt">Table of n, a(n) for n = 0..133</a>
%H A322529 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A322529.py">Python program</a>.
%e A322529 The a(30) = 8 integer partitions:
%e A322529   (30),
%e A322529   (17,13),(19,11),(23,7),
%e A322529   (17,11,2),(23,5,2),
%e A322529   (13,7,5,3,2),
%e A322529   (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1).
%t A322529 Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SquareFreeQ[Times@@#]]&]],{n,30}]
%Y A322529 Cf. A002865, A003963, A005117, A038041, A064573, A073576, A302505, A319056, A320322, A321717, A321718, A322526, A322527, A322528, A322530, A322531.
%K A322529 nonn
%O A322529 0,3
%A A322529 _Gus Wiseman_, Dec 14 2018
%E A322529 a(51)-a(69) from _Jinyuan Wang_, Jun 27 2020
%E A322529 a(70) onwards from _Lucas A. Brown_, Aug 17 2024