cp's OEIS Frontend

A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

%I A387180 #7 Aug 31 2025 10:49:08
%S A387180 4,8,12,16,20,24,27,28,32,36,40,44,48,52,54,56,60,64,68,72,76,80,81,
%T A387180 84,88,92,96,100,104,108,112,116,120,124,125,128,132,135,136,140,144,
%U A387180 148,152,156,160,162,164,168,172,176,180,184,188,189,192,196,200,204
%N A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.
%C A387180 First differs from A276079 in having 125.
%C A387180 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A387180 Also numbers n with at least one prime index k such that the multiplicity of prime(k) in the prime factorization of n exceeds the number of divisors of k.
%e A387180 The prime indices of 60 are {1,1,2,3}, and we have the following 4 choices of constant partitions:
%e A387180   ((1),(1),(2),(3))
%e A387180   ((1),(1),(2),(1,1,1))
%e A387180   ((1),(1),(1,1),(3))
%e A387180   ((1),(1),(1,1),(1,1,1))
%e A387180 Since none of these is strict, 60 is in the sequence.
%e A387180 The prime indices of 90 are {1,2,2,3}, and we have the following 4 strict choices:
%e A387180   ((1),(2),(1,1),(3))
%e A387180   ((1),(2),(1,1),(1,1,1))
%e A387180   ((1),(1,1),(2),(3))
%e A387180   ((1),(1,1),(2),(1,1,1))
%e A387180 So 90 is not in the sequence.
%t A387180 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387180 Select[Range[100],Select[Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[#]],UnsameQ@@#&]=={}&]
%Y A387180 For prime factors instead of constant partitions we have A355529, counted by A370593.
%Y A387180 For divisors instead of constant partitions we have A355740, counted by A370320.
%Y A387180 The complement for prime factors is A368100, counted by A370592.
%Y A387180 The complement for divisors is A368110, counted by A239312.
%Y A387180 The complement for initial intervals is A387112, counted by A238873.
%Y A387180 For initial intervals instead of partitions we have A387113, counted by A387118.
%Y A387180 These are the positions of zero in A387120.
%Y A387180 For strict instead of constant partitions we have A387176, counted by A387137.
%Y A387180 The complement for strict partitions is A387177, counted by A387178.
%Y A387180 Twice-partitions of this type are counted by A387179, constant-block case of A296122.
%Y A387180 The complement is A387181 (nonzeros of A387120), counted by A387330.
%Y A387180 Partitions of this type are counted by A387329.
%Y A387180 A000041 counts integer partitions, strict A000009.
%Y A387180 A003963 multiplies together prime indices.
%Y A387180 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387180 A120383 lists numbers divisible by all of their prime indices.
%Y A387180 A289509 lists numbers with relatively prime prime indices.
%Y A387180 Cf. A355739, A367771, A387111, A387115.
%Y A387180 Cf. A000005, A052335, A063834, A276079, A299200, A299201, A335433, A335448, A355731, A383706, A387110.
%K A387180 nonn,new
%O A387180 1,1
%A A387180 _Gus Wiseman_, Aug 30 2025