A347060 -id:A347060 - cp's OEIS frontend

A066624 Number of 1's in binary expansion of parts in all partitions of n.

0, 1, 3, 7, 13, 23, 41, 65, 102, 156, 234, 340, 495, 697, 982, 1359, 1864, 2523, 3408, 4536, 6022, 7918, 10365, 13457, 17423, 22380, 28666, 36498, 46318, 58466, 73617, 92221, 115236, 143402, 177984, 220086, 271524, 333810, 409490, 500804, 611149, 743728, 903296

Offset: 0

Naohiro Nomoto, Jan 09 2002

			For n = 3: 11 = 10+1 = 1+1+1 [binary expansion of partitions of 3]. a(3) = (two 1's) + (two 1's) + (three 1's), so a(3) = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..8000

Cf. A000120, A000070, A347060.

<< DiscreteMath`Combinatorica`; Table[Count[Flatten[IntegerDigits[Partitions[n], 2]], 1], {n, 0, 50}]
Table[Total[Flatten[IntegerDigits[#,2]&/@IntegerPartitions[n]]],{n,0,50}] (* Harvey P. Dale, Mar 29 2022 *)

More terms from Vladeta Jovovic and Robert G. Wilson v, Jan 11 2002

A319140 Total number of binary digits in all partitions of n into distinct parts.

Original entry on oeis.org

1, 2, 5, 6, 11, 17, 23, 30, 44, 60, 76, 102, 128, 166, 214, 264, 327, 413, 502, 618, 759, 917, 1105, 1335, 1598, 1907, 2279, 2702, 3191, 3776, 4436, 5198, 6101, 7113, 8292, 9653, 11188, 12951, 14984, 17277, 19889, 22881, 26248, 30073, 34439, 39320, 44850

Offset: 1

David S. Newman, Sep 11 2018

			For n = 4 there are 2 partitions into distinct parts in binary they are: 100, 11+1, for a total of 6 binary parts.

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

Cf. A000009, A070939, A318756, A319142, A347060.

h:= proc(n) option remember; 1+ilog2(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+h(i)
       *[0, p[1]])(b(n-i, min(n-i, i-1)))))
    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[n > i*(i+1)/2, 0, b[n, i-1] + Function[p, p + h[i]*{0, p[[1]]}][b[n-i, Min[n-i, i-1]]]]];
a[n_] := b[n, n][[2]];
a /@ Range[1; 60] (* Jean-François Alcover, Sep 28 2019, after Alois P. Heinz *)

seq(n)={[subst(deriv(p,y), y, 1) | p<-Vec(-1 + prod(k=1, n, 1 + x^k*y^(logint(k,2)+1) + O(x*x^n)))]} \\ Andrew Howroyd, Sep 17 2018

cp's OEIS Frontend

A066624 Number of 1's in binary expansion of parts in all partitions of n.

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