cp's OEIS Frontend

A376657 Number of integer factorizations of n into nonsquarefree factors > 1.

%I A376657 #10 Oct 08 2024 18:39:44
%S A376657 1,0,0,1,0,0,0,1,1,0,0,1,0,0,0,2,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,2,0,0,
%T A376657 0,2,0,0,0,1,0,0,0,1,1,0,0,2,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,4,0,0,0,1,
%U A376657 0,0,0,3,0,0,1,1,0,0,0,2,2,0,0,1,0,0,0
%N A376657 Number of integer factorizations of n into nonsquarefree factors > 1.
%e A376657 The a(n) factorizations for n = 16, 64, 72, 144, 192, 256, 288:
%e A376657   (16)   (64)     (72)    (144)    (192)     (256)      (288)
%e A376657   (4*4)  (8*8)    (8*9)   (4*36)   (4*48)    (4*64)     (4*72)
%e A376657          (4*16)   (4*18)  (8*18)   (8*24)    (8*32)     (8*36)
%e A376657          (4*4*4)          (9*16)   (12*16)   (16*16)    (9*32)
%e A376657                           (12*12)  (4*4*12)  (4*8*8)    (12*24)
%e A376657                           (4*4*9)            (4*4*16)   (16*18)
%e A376657                                              (4*4*4*4)  (4*8*9)
%e A376657                                                         (4*4*18)
%t A376657 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A376657 Table[Length[Select[facs[n],NoneTrue[SquareFreeQ]]],{n,100}]
%Y A376657 For prime-powers we have A000688.
%Y A376657 Positions of zeros are A005117 (squarefree numbers), complement A013929.
%Y A376657 For squarefree instead of nonsquarefree we have A050320, strict A050326.
%Y A376657 For nonprime numbers we have A050370.
%Y A376657 The version for partitions is A114374.
%Y A376657 For perfect-powers we have A294068.
%Y A376657 For non-perfect-powers we have A303707.
%Y A376657 For non-prime-powers we have A322452.
%Y A376657 The strict case is A376679.
%Y A376657 Nonsquarefree numbers:
%Y A376657 - A078147 (first differences)
%Y A376657 - A376593 (second differences)
%Y A376657 - A376594 (inflections and undulations)
%Y A376657 - A376595 (nonzero curvature)
%Y A376657 A000040 lists the prime numbers, differences A001223.
%Y A376657 A001055 counts integer factorizations, strict A045778.
%Y A376657 A005117 lists squarefree numbers, differences A076259.
%Y A376657 A317829 counts factorizations of superprimorials, strict A337069.
%Y A376657 Cf. A008480, A053797, A053806, A061398, A089259, A120992, A124010, A182853, A373198, A375707, A376312.
%K A376657 nonn
%O A376657 1,16
%A A376657 _Gus Wiseman_, Oct 07 2024