A330996 -id:A330996 - cp's OEIS frontend

A331022 Numbers k such that the number of strict integer partitions of k is a power of 2.

0, 1, 2, 3, 4, 6, 9, 16, 20, 29, 34, 45

Offset: 0

Gus Wiseman, Jan 10 2020

An integer partition of n is a finite, nonincreasing sequence of positive integers (parts) summing to n. It is strict if the parts are all different. Integer partitions and strict integer partitions are counted by A000041 and A000009 respectively.
Conjecture: This sequence is finite.
Conjecture: The analogous sequence for non-strict partitions is: 0, 1, 2.
Next term > 5*10^4 if it exists. - Seiichi Manyama, Jan 12 2020

			The strict integer partitions of the initial terms:
  (1)  (2)  (3)    (4)    (6)      (9)
            (2,1)  (3,1)  (4,2)    (5,4)
                          (5,1)    (6,3)
                          (3,2,1)  (7,2)
                                   (8,1)
                                   (4,3,2)
                                   (5,3,1)
                                   (6,2,1)

The version for primes instead of powers of 2 is A035359.
The version for factorizations instead of strict partitions is A330977.
Numbers whose number of partitions is prime are A046063.
Cf. A000009, A000079, A001055, A036498, A045778, A318286, A330989, A330990, A330994/A330995, A330996.

Select[Range[0,1000],IntegerQ[Log[2,PartitionsQ[#]]]&]

A330994 Numerator of P(n)/Q(n) = A000041(n)/A000009(n).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 3, 11, 15, 21, 14, 77, 101, 135, 176, 231, 297, 385, 245, 627, 198, 1002, 1255, 1575, 979, 812, 1505, 1859, 4565, 1401, 3421, 2783, 1449, 6155, 4961, 17977, 21637, 26015, 31185, 1778, 2123, 26587, 63261, 75175, 44567, 17593, 8911, 49091

Offset: 0

Gus Wiseman, Jan 08 2020

An integer partition of n is a finite, nonincreasing sequence of positive integers (parts) summing to n. It is strict if the parts are all different. Integer partitions and strict integer partitions are counted by A000041 and A000009 respectively.

The denominators are A330995.
The rounded quotients are A330996.
The same for factorizations is A331023.
Cf. A000009, A000041, A003238, A005117, A035359, A046063, A331022.

Table[PartitionsP[n]/PartitionsQ[n],{n,0,100}]//Numerator

A330994/A330995 = A000041/A000009.

A330995 Denominator P(n)/Q(n) = A000041(n)/A000009(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 1, 3, 4, 5, 3, 15, 18, 22, 27, 32, 38, 46, 27, 64, 19, 89, 104, 122, 71, 55, 96, 111, 256, 74, 170, 130, 64, 256, 195, 668, 760, 864, 982, 53, 60, 713, 1610, 1816, 1024, 384, 185, 970, 3264, 1829, 4097, 4582, 5120, 5718, 3189, 7108, 2639

Offset: 0

Gus Wiseman, Jan 08 2020

An integer partition of n is a finite, nonincreasing sequence of positive integers (parts) summing to n. It is strict if the parts are all different. Integer partitions and strict integer partitions are counted by A000041 and A000009 respectively.

Conjecture: The only 1's occur at n = 0, 1, 2, 7.

The numerators are A330994.
The rounded quotients are A330996.
The same for factorizations is A331024.
Cf. A000009, A000041, A001055, A003238, A005117, A035359, A045778, A046063, A331022.

Table[PartitionsP[n]/PartitionsQ[n],{n,0,100}]//Denominator

A330994/A330995 = A000041/A000009.

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

Original entry on oeis.org

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}]

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A331022 Numbers k such that the number of strict integer partitions of k is a power of 2.

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A330994 Numerator of P(n)/Q(n) = A000041(n)/A000009(n).

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A330995 Denominator P(n)/Q(n) = A000041(n)/A000009(n).

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

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