A070940 -id:A070940 - cp's OEIS frontend

A307313 a(n) is the denominator of n/2^(length of the binary representation of n).

2, 2, 4, 2, 8, 4, 8, 2, 16, 8, 16, 4, 16, 8, 16, 2, 32, 16, 32, 8, 32, 16, 32, 4, 32, 16, 32, 8, 32, 16, 32, 2, 64, 32, 64, 16, 64, 32, 64, 8, 64, 32, 64, 16, 64, 32, 64, 4, 64, 32, 64, 16, 64, 32, 64, 8, 64, 32, 64, 16, 64, 32, 64, 2, 128, 64, 128, 32, 128, 64

Offset: 1

Michel Marcus, Apr 02 2019

			For n=1, 1 = 1_2,  a(1) = denominator(1/(2^1)) = denominator(1/2) = 2;
For n=2, 2 = 10_2, a(2) = denominator(2/(2^2)) = denominator(1/2) = 2;
For n=3, 3 = 11_2, a(3) = denominator(3/(2^2)) = denominator(3/4) = 4.

Cf. A062383, A070939, A000265 (numerators), A078267 (analog in base 10).

PARI
```
a(n) = denominator(n/(2^(#binary(n))));
```

a(n) = denominator(n/2^A070939(n)).
a(n) = denominator(n/A062383(n)).
a(n) = 2^A070940(n).

A373120 Number of distinct possible binary ranks of integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 33, 43, 55, 70, 89, 109, 136, 167, 206, 251, 306, 371, 445, 535, 639, 759, 904, 1069, 1262, 1489, 1747, 2047, 2390, 2784, 3237, 3754, 4350, 5027, 5798, 6680, 7671, 8808, 10091, 11543, 13190, 15040, 17128, 19477, 22118

Offset: 0

Gus Wiseman, May 26 2024

nonn

			The partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1), with respective binary ranks 8, 5, 4, 4, 4, so a(4) = 3.

The strict case is A000009.
A048675 gives binary rank of prime indices, distinct A087207.
A118462 lists binary ranks of strict integer partitions, row sums A372888.
A277905 groups all positive integers by binary rank of prime indices.
A372890 adds up binary ranks of integer partitions.
Binary indices (A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- max A029837 or A070939, opposite A070940
- sum A029931, product A096111
- reverse A272020
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000041, A018819, A158704, A158705, A225620, A231204, A372688.

Table[Length[Union[Total[2^(#-1)]&/@IntegerPartitions[n]]],{n,0,15}]

cp's OEIS Frontend

A307313 a(n) is the denominator of n/2^(length of the binary representation of n).

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