A066624 -id:A066624 - cp's OEIS frontend

A318756 Total number of binary digits used to write all partitions of n in binary notation.

1, 4, 8, 18, 30, 55, 85, 141, 211, 324, 467, 691, 968, 1377, 1898, 2631, 3554, 4830, 6425, 8578, 11272, 14819, 19243, 25005, 32133, 41279, 52585, 66907, 84512, 106636, 133685, 167377, 208439, 259145, 320696, 396251, 487532, 598881, 732990, 895627, 1090752

Offset: 1

David S. Newman, Sep 02 2018

			For n = 3 there are 3 partitions which when written in binary are: 11, 10+1, 1+1+1, for a total of 8 binary integers.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A066624, A070939.

h:= proc(n) option remember; 1+ilog2(n) end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, b(n, i-1)+(p-> p+[0, p[1]*
        h(i)])(b(n-i, min(n-i, i)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 27 2018

h[n_] := h[n] = 1 + Log[2, n] // Floor;
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, 0, b[n, i - 1] + Function[p, p + {0, p[[1]]*h[i]}][b[n - i, Min[n - i, i]]]]];
a[n_] := b[n, n][[2]];
a /@ Range[1, 60] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz *)
Table[Length[Flatten[IntegerDigits[#,2]&/@IntegerPartitions[n]]],{n,50}] (* Harvey P. Dale, Aug 14 2021 *)

a(n)={subst(deriv(polcoef(1/prod(k=1, n, 1 - x^k*y^(logint(k,2) + 1) + O(x*x^n)), n)), y, 1)} \\ Andrew Howroyd, Sep 07 2018

A347060 Total number of 1's in the binary expansion of parts in all partitions of n into distinct parts.

Original entry on oeis.org

0, 1, 1, 4, 4, 7, 11, 15, 20, 28, 39, 48, 64, 80, 104, 134, 167, 203, 257, 311, 381, 470, 566, 680, 820, 981, 1168, 1394, 1650, 1946, 2300, 2700, 3161, 3705, 4315, 5026, 5845, 6769, 7827, 9049, 10424, 11992, 13784, 15801, 18088, 20702, 23620, 26922, 30665

Offset: 0

Alois P. Heinz, Aug 14 2021

			a(5) = 7 counts the 1's in [101], [100, 1], [11, 10].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000009, A000120, A066189, A066624, A318756, A319140.

h:= proc(n) option remember; add(i, i=Bits[Split](n)) end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(n>i*(i+1)/2, 0, b(n, i-1)+(p-> p+
       [0, p[1]*h(i)])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);

cp's OEIS Frontend

A318756 Total number of binary digits used to write all partitions of n in binary notation.

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