A372890 Sum of binary ranks of all integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

0, 1, 4, 10, 25, 52, 115, 228, 471, 931, 1871, 3687, 7373, 14572, 29049, 57694, 115058, 229101, 457392, 912469, 1822945, 3640998, 7277426, 14544436, 29079423, 58137188, 116254386, 232465342, 464889800, 929691662, 1859302291, 3718428513, 7436694889, 14873042016

Offset: 0

Author

Gus Wiseman, May 23 2024

Keywords

• nonn

Examples

			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 with sum 25, so a(4) = 25.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3321

Crossrefs

For Heinz number (not binary rank) we have A145519, row sums of A215366.

For Heinz number the strict version is A147655, row sums of A246867.

The strict version is A372888, row sums of A118462.

A005117 gives Heinz numbers of strict integer partitions.

A048675 gives binary rank of prime indices, distinct A087207.

A061395 gives greatest prime index, least A055396.

A118457 lists strict partitions in Mathematica order.

A277905 groups all positive integers by binary rank of prime indices.

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, A005940, A018819, A019565, A066633, A225620, A344086, A365410.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
      b(n, i-1)+(p->[0, p[1]*2^(i-1)]+p)(b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..33);  # Alois P. Heinz, May 23 2024

Mathematica

Table[Total[Total[2^(#-1)]&/@IntegerPartitions[n]],{n,0,10}]

Formula

From Alois P. Heinz, May 23 2024: (Start)

a(n) = Sum_{k=1..n} 2^(k-1) * A066633(n,k).

a(n) mod 2 = A365410(n-1) for n>=1. (End)