A323087 -id:A323087 - cp's OEIS frontend

A323054 Number of strict integer partitions of n with no 1's such that no part is a power of any other part.

1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 4, 6, 6, 8, 9, 12, 13, 16, 19, 21, 25, 30, 36, 40, 47, 53, 63, 71, 83, 94, 107, 121, 140, 159, 180, 204, 233, 260, 296, 334, 377, 421, 474, 532, 598, 668, 750, 835, 933, 1038, 1163, 1292, 1435, 1597, 1771, 1966, 2180, 2421, 2673

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(2) = 1 through a(13) = 8 strict integer partitions (A = 10, B = 11, C = 12, D = 13):
  (2)  (3)  (4)  (5)   (6)  (7)   (8)   (9)   (A)    (B)    (C)    (D)
                 (32)       (43)  (53)  (54)  (64)   (65)   (75)   (76)
                            (52)  (62)  (63)  (73)   (74)   (84)   (85)
                                        (72)  (532)  (83)   (A2)   (94)
                                                     (92)   (543)  (A3)
                                                     (632)  (732)  (B2)
                                                                   (643)
                                                                   (652)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

Cf. A001597, A007916, A025147, A052410, A078374, A087897, A120641, A275972, A303362, A323053, A323087, A323088.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],UnsameQ@@#,stableQ[#,IntegerQ[Log[#1,#2]]&]]&]],{n,30}]

A323090 Number of strict factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 2, 0, 2, 0, 2, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 0, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 7, 1, 2, 2, 0, 2, 5, 1, 2, 2, 5, 1, 4, 1, 2, 2, 2, 2, 5, 1, 2, 0, 2, 1, 7, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

			The a(72) = 4 factorizations are (2*3*12), (3*24), (6*12), (72). Missing from this list and not strict are (2*2*2*3*3), (2*2*3*6), (2*6*6), (2*2*18), while missing from the list and using perfect powers are (2*36), (2*4*9), (3*4*6), (4*18), (8*9).

Positions of 0's are A246547.
Positions of 1's are A000040.
Positions of 2's are A084227.
Positions of 3's are A085986.
Positions of 4's are A143610.
Cf. A001055, A001597, A007916, A025147, A045778, A052410, A303707, A323054, A323087, A323088, A323089.

radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
facssr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facssr[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[facssr[n]],{n,100}]

A323086 Number of factorizations of n into factors > 1 such that no factor is a power of any other (unequal) factor.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 3, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 9, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 4, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 9, 3, 2, 1, 11, 2, 2

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

			The a(72) = 14 factorizations:
  (2*2*2*3*3),
  (2*2*2*9), (2*2*3*6),
  (2*2*18), (2*3*12), (2*6*6), (3*3*8), (3*4*6),
  (2*36), (3*24), (4*18), (6*12), (8*9),
  (72).

Cf. A001055, A001597, A002865, A007916, A052410, A101417, A292886, A303707, A305149, A323053, A323087.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[facs[n],stableQ[Union[#],IntegerQ[Log[#1,#2]]&]&]],{n,100}]

A323091 Number of strict knapsack factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 4, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 3, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

A strict knapsack factorization is a finite set of positive integers > 1 such that every subset has a different product.

			The a(144) = 11 factorizations:
  (144),
  (2*72), (3*48), (4*36),(6*24), (8*18), (9*16),
  (2*3*24), (2*4*18), (2*8*9), (3*6*8).
Missing from this list are (2*6*12), (3*4*12), (2*3*4*6), which are not knapsack.

Cf. A001055, A045778, A108917, A120641, A275972, A292886, A305150, A323087.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[strfacs[n],UnsameQ@@Times@@@Subsets[#]&]],{n,100}]

a(prime^n) = A275972(n).

cp's OEIS Frontend

A323054 Number of strict integer partitions of n with no 1's such that no part is a power of any other part.

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A323090 Number of strict factorizations of n using elements of A007916 (numbers that are not perfect powers).

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A323086 Number of factorizations of n into factors > 1 such that no factor is a power of any other (unequal) factor.

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A323091 Number of strict knapsack factorizations of n.

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