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A376657
Number of integer factorizations of n into nonsquarefree factors > 1.
1
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, 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, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0
OFFSET
1,16
EXAMPLE
The a(n) factorizations for n = 16, 64, 72, 144, 192, 256, 288:
(16) (64) (72) (144) (192) (256) (288)
(4*4) (8*8) (8*9) (4*36) (4*48) (4*64) (4*72)
(4*16) (4*18) (8*18) (8*24) (8*32) (8*36)
(4*4*4) (9*16) (12*16) (16*16) (9*32)
(12*12) (4*4*12) (4*8*8) (12*24)
(4*4*9) (4*4*16) (16*18)
(4*4*4*4) (4*8*9)
(4*4*18)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], NoneTrue[SquareFreeQ]]], {n, 100}]
CROSSREFS
For prime-powers we have A000688.
Positions of zeros are A005117 (squarefree numbers), complement A013929.
For squarefree instead of nonsquarefree we have A050320, strict A050326.
For nonprime numbers we have A050370.
The version for partitions is A114374.
For perfect-powers we have A294068.
For non-perfect-powers we have A303707.
For non-prime-powers we have A322452.
The strict case is A376679.
Nonsquarefree numbers:
- A078147 (first differences)
- A376593 (second differences)
- A376594 (inflections and undulations)
- A376595 (nonzero curvature)
A000040 lists the prime numbers, differences A001223.
A001055 counts integer factorizations, strict A045778.
A005117 lists squarefree numbers, differences A076259.
A317829 counts factorizations of superprimorials, strict A337069.
Sequence in context: A121467 A366073 A362412 * A071325 A064727 A343222
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 07 2024
STATUS
approved