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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).

%I A372890 #15 May 24 2024 14:30:29
%S A372890 0,1,4,10,25,52,115,228,471,931,1871,3687,7373,14572,29049,57694,
%T A372890 115058,229101,457392,912469,1822945,3640998,7277426,14544436,
%U A372890 29079423,58137188,116254386,232465342,464889800,929691662,1859302291,3718428513,7436694889,14873042016
%N 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).
%H A372890 Alois P. Heinz, <a href="/A372890/b372890.txt">Table of n, a(n) for n = 0..3321</a>
%F A372890 From _Alois P. Heinz_, May 23 2024: (Start)
%F A372890 a(n) = Sum_{k=1..n} 2^(k-1) * A066633(n,k).
%F A372890 a(n) mod 2 = A365410(n-1) for n>=1. (End)
%e A372890 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.
%p A372890 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
%p A372890       b(n, i-1)+(p->[0, p[1]*2^(i-1)]+p)(b(n-i, min(n-i, i))))
%p A372890     end:
%p A372890 a:= n-> b(n$2)[2]:
%p A372890 seq(a(n), n=0..33);  # _Alois P. Heinz_, May 23 2024
%t A372890 Table[Total[Total[2^(#-1)]&/@IntegerPartitions[n]],{n,0,10}]
%Y A372890 For Heinz number (not binary rank) we have A145519, row sums of A215366.
%Y A372890 For Heinz number the strict version is A147655, row sums of A246867.
%Y A372890 The strict version is A372888, row sums of A118462.
%Y A372890 A005117 gives Heinz numbers of strict integer partitions.
%Y A372890 A048675 gives binary rank of prime indices, distinct A087207.
%Y A372890 A061395 gives greatest prime index, least A055396.
%Y A372890 A118457 lists strict partitions in Mathematica order.
%Y A372890 A277905 groups all positive integers by binary rank of prime indices.
%Y A372890 Binary indices (A048793):
%Y A372890 - length A000120, complement A023416
%Y A372890 - min A001511, opposite A000012
%Y A372890 - max A029837 or A070939, opposite A070940
%Y A372890 - sum A029931, product A096111
%Y A372890 - reverse A272020
%Y A372890 - complement A368494, sum A359400
%Y A372890 - opposite complement A371571, sum A359359
%Y A372890 - opposite A371572, sum A230877
%Y A372890 Cf. A000041, A005940, A018819, A019565, A066633, A225620, A344086, A365410.
%K A372890 nonn
%O A372890 0,3
%A A372890 _Gus Wiseman_, May 23 2024