A045780 -id:A045780 - cp's OEIS frontend

A331048 Nearest integer to A001055(n)/A045778(n), where A001055 is factorizations and A045778 is strict factorizations.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jan 10 2020

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. It is strict if the factors are all different.

The exact quotient is A331023/A331024.
The same for integer partitions is A330996 ~ A330994/A330995.
Cf. A001055, A001222, A002033, A005117, A045778, A045779, A045780, A045782, A045783, A330972, A330977, A330991.

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

A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A080257 in lacking 60.

			Strict factorizations of selected terms:
  (6)    (12)   (24)     (48)     (216)
  (2*3)  (2*6)  (3*8)    (6*8)    (3*72)
         (3*4)  (4*6)    (2*24)   (4*54)
                (2*12)   (3*16)   (6*36)
                (2*3*4)  (4*12)   (8*27)
                         (2*3*8)  (9*24)
                         (2*4*6)  (12*18)
                                  (2*108)
                                  (3*8*9)
                                  (4*6*9)
                                  (2*3*36)
                                  (2*4*27)
                                  (2*6*18)
                                  (2*9*12)
                                  (3*4*18)
                                  (3*6*12)
                                  (2*3*4*9)

The version for strict integer partitions is A035359.
The version for integer partitions is A046063.
The version for set partitions is A051130.
The non-strict version is A330991.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
Numbers whose number of strict factorizations is odd are A331230.
Numbers whose number of strict factorizations is even are A331231.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A318286, A328966, A330992, A330993, A330997, A331023/A331024, A331050, A331051, A331200, A331232.

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

A331230 Numbers k such that the number of factorizations of k into distinct factors > 1 is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 24, 25, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 83, 84, 88, 89, 90, 92, 97, 98, 99, 100, 101, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A319237 in lacking 300.

The version for strict integer partitions is A001318.
The version for integer partitions is A052002.
The version for set partitions appears to be A032766.
The non-strict version is A331050.
The version for primes (instead of odds) is A331201.
The even version is A331231.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
Cf. A035359, A045780, A318286, A328966, A330991, A330997, A331051, A331200, A331232.

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

A331231 Numbers k such that the number of factorizations of k into distinct factors > 1 is even.

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 160, 161, 166

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A319238 in having 300.

The version for integer partitions is A001560.
The version for strict integer partitions is A090864.
The version for set partitions appears to be A016789.
The non-strict version is A331051.
The version for primes (instead of evens) is A331201.
The odd version is A331230.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A045783, A318286, A328966, A330973, A330991, A330997, A331022, A331023/A331024, A331050, A331200, A331232.

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

A331198 Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).

Original entry on oeis.org

128, 2187, 10368, 34992, 78125, 80000, 307328, 823543, 1250000, 1366875, 1874048, 3655808, 5250987, 6328125, 10690688, 13176688, 16681088, 19487171, 32019867, 35819648, 62462907, 62748517, 66706983, 90531968, 118210688, 182660427, 187578125, 239892608, 285012027

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

Contains p^7 for all primes p.

			The 15 factorizations and 5 strict factorizations of 2187:
  (2187)           (2187)
  (27*81)          (27*81)
  (3*729)          (3*729)
  (9*243)          (9*243)
  (3*9*81)         (3*9*81)
  (9*9*27)
  (3*27*27)
  (3*3*243)
  (3*9*9*9)
  (3*3*3*81)
  (3*3*9*27)
  (3*3*3*9*9)
  (3*3*3*3*27)
  (3*3*3*3*3*9)
  (3*3*3*3*3*3*3)

Factorizations are A001055.
Strict factorizations are A045778.
Taking "twice" instead of "three times" gives A001248.
Cf. A045779, A045780, A045782, A045783, A330972, A331023/A331024, A331048, A331050.

facsm[n_]:=facsm[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsm[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100000],3==Length[facsm[#]]/Length[Select[facsm[#],UnsameQ@@#&]]&]

a(7)-(10) from Alois P. Heinz, Jan 17 2020

a(11)-a(29) from Giovanni Resta, Jan 20 2020

cp's OEIS Frontend

A331048 Nearest integer to A001055(n)/A045778(n), where A001055 is factorizations and A045778 is strict factorizations.

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A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

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A331230 Numbers k such that the number of factorizations of k into distinct factors > 1 is odd.

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A331231 Numbers k such that the number of factorizations of k into distinct factors > 1 is even.

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A331198 Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).

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