A231204 -id:A231204 - cp's OEIS frontend

A018819 Binary partition function: number of partitions of n into powers of 2.

1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, 26, 26, 36, 36, 46, 46, 60, 60, 74, 74, 94, 94, 114, 114, 140, 140, 166, 166, 202, 202, 238, 238, 284, 284, 330, 330, 390, 390, 450, 450, 524, 524, 598, 598, 692, 692, 786, 786, 900, 900, 1014, 1014, 1154, 1154, 1294, 1294

Offset: 0

David W. Wilson, N. J. A. Sloane and J. H. Conway

First differences of A000123; also A000123 with terms repeated. See the relevant proof that follows the first formula below.
Among these partitions there is exactly one partition with all distinct terms, as every number can be expressed as the sum of the distinct powers of 2.
Euler transform of A036987 with offset 1.
a(n) is the number of "non-squashing" partitions of n, that is, partitions n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k and p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k. - N. J. A. Sloane, Nov 30 2003
Normally the OEIS does not include sequences like this where every term is repeated, but an exception was made for this one because of its importance. The unrepeated sequence A000123 is the main entry.
Number of different partial sums from 1 + [1, *2] + [1, *2] + ..., where [1, *2] means we can either add 1 or multiply by 2. E.g., a(6) = 6 because we have 6 = 1 + 1 + 1 + 1 + 1 + 1 = (1+1) * 2 + 1 + 1 = 1 * 2 * 2 + 1 + 1 = (1+1+1) * 2 = 1 * 2 + 1 + 1 + 1 + 1 = (1*2+1) * 2 where the connection is defined via expanding each bracket; e.g., this is 6 = 1 + 1 + 1 + 1 + 1 + 1 = 2 + 2 + 1 + 1 = 4 + 1 + 1 = 2 + 2 + 2 = 2 + 1 + 1 + 1 + 1 = 4 + 2. - Jon Perry, Jan 01 2004
Number of partitions p of n such that the number of compositions generated by p is odd. For proof see the Alekseyev and Adams-Watters link. - Vladeta Jovovic, Aug 06 2007
Differs from A008645 first at a(64). - R. J. Mathar, May 28 2008
Appears to be row sums of A155077. - Mats Granvik, Jan 19 2009
Number of partitions (p_1, p_2, ..., p_k) of n, with p_1 >= p_2 >= ... >= p_k, such that for each i, p_i >= p_{i+1} + ... + p_k. - John MCKAY (mckay(AT)encs.concordia.ca), Mar 06 2009 (these are the "non-squashing" partitions as nonincreasing lists).
Equals rightmost diagonal of triangle of A168261. Starting with offset 1 = eigensequence of triangle A115361 and row sums of triangle A168261. - Gary W. Adamson, Nov 21 2009
Equals convolution square root of A171238: (1, 2, 5, 8, 16, 24, 40, 56, 88, ...). - Gary W. Adamson, Dec 05 2009
Let B = the n-th convolution power of the sequence and C = the aerated variant of B. It appears that B/C = the binomial sequence beginning (1, n, ...). Example: Third convolution power of the sequence is (1, 3, 9, 19, 42, 78, 146, ...), with C = (1, 0, 3, 0, 9, 0, 19, ...). Then B/C = (1, 3, 6, 10, 15, 21, ...). - Gary W. Adamson, Aug 15 2016
From Gary W. Adamson, Sep 08 2016: (Start)
The limit of the matrix power M^k as n-->inf results in a single column vector equal to the sequence, where M is the following production matrix:
  1, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, ...
  1, 1, 0, 0, 0, ...
  1, 1, 1, 0, 0, ...
  1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, ...
  ... (End)
a(n) is the number of "non-borrowing" partitions of n, meaning binary subtraction of a smaller part from a larger part will never require place-value borrowing. - David V. Feldman, Jan 29 2020
From Gus Wiseman, May 25 2024: (Start)
Also the number of multisets of positive integers whose binary rank is n, where the binary rank of a multiset m is given by Sum_i 2^(m_i-1). For example, the a(1) = 1 through a(8) = 10 multisets are:
  {1}  {2}   {12}   {3}     {13}     {23}      {123}      {4}
       {11}  {111}  {22}    {122}    {113}     {1113}     {33}
                    {112}   {1112}   {222}     {1222}     {223}
                    {1111}  {11111}  {1122}    {11122}    {1123}
                                     {11112}   {111112}   {2222}
                                     {111111}  {1111111}  {11113}
                                                          {11222}
                                                          {111122}
                                                          {1111112}
                                                          {11111111}
(End)

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 6*x^7 + 10*x^8 + ...
a(4) = 4: the partitions are 4, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1.
a(7) = 6: the partitions are 4 + 2 + 1, 4 + 1 + 1 + 1, 2 + 2 + 2 + 1, 2 + 2 + 1 + 1 + 1, 2 + 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 + 1 + 1.
From _Joerg Arndt_, Dec 17 2012: (Start)
The a(10) = 14 binary partitions of 10 are (in lexicographic order)
[ 1]  [ 1 1 1 1 1 1 1 1 1 1 ]
[ 2]  [ 2 1 1 1 1 1 1 1 1 ]
[ 3]  [ 2 2 1 1 1 1 1 1 ]
[ 4]  [ 2 2 2 1 1 1 1 ]
[ 5]  [ 2 2 2 2 1 1 ]
[ 6]  [ 2 2 2 2 2 ]
[ 7]  [ 4 1 1 1 1 1 1 ]
[ 8]  [ 4 2 1 1 1 1 ]
[ 9]  [ 4 2 2 1 1 ]
[10]  [ 4 2 2 2 ]
[11]  [ 4 4 1 1 ]
[12]  [ 4 4 2 ]
[13]  [ 8 1 1 ]
[14]  [ 8 2 ]
The a(11) = 14 binary partitions of 11 are obtained by appending 1 to each partition in the list.
The a(10) = 14 non-squashing partitions of 10 are (in lexicographic order)
[ 1]  [ 6 3 1 1 ]
[ 2]  [ 6 3 2 ]
[ 3]  [ 6 4 1 ]
[ 4]  [ 6 5 ]
[ 5]  [ 7 2 1 1 ]
[ 6]  [ 7 2 2 ]
[ 7]  [ 7 3 1 ]
[ 8]  [ 7 4 ]
[ 9]  [ 8 2 1 ]
[10]  [ 8 3 ]
[11]  [ 9 1 1 ]
[12]  [ 9 2 ]
[13]  [ 10 1 ]
[14]  [ 11 ]
The a(11) = 14 non-squashing partitions of 11 are obtained by adding 1 to the first part in each partition in the list.
(End)
From _David V. Feldman_, Jan 29 2020: (Start)
The a(10) = 14 non-borrowing partitions of 10 are (in lexicographic order)
[ 1] [1 1 1 1 1 1 1 1 1 1]
[ 2] [2 2 2 2 2]
[ 3] [3 1 1 1 1 1 1 1]
[ 4] [3 3 1 1 1 1]
[ 5] [3 3 2 2]
[ 6] [3 3 3 1]
[ 7] [5 1 1 1 1 1]
[ 8] [5 5]
[ 9] [6 2 2]
[10] [6 4]
[11] [7 1 1 1]
[12] [7 3]
[13] [9 1]
[14] [10]
The a(11) = 14 non-borrowing partitions of 11 are obtained either by adding 1 to the first even part in each partition (if any) or else appending a 1 after the last part.
(End)
For example, the five partitions of 4, written in nonincreasing order, are [1, 1, 1, 1], [2, 1, 1], [2, 2], [3, 1], [4]. The last four satisfy the condition, and a(4) = 4. The Maple program below verifies this for small values of n.

T. D. Noe, Table of n, a(n) for n = 0..1000
Max Alekseyev and Franklin T. Adams-Watters, Two proofs of an observation of Vladeta Jovovic
Giedrius Alkauskas, Generalization of the Rodseth-Gupta theorem on binary partitions, Lithuanian Math. J., 43 (2) (2003), 103-110.
Giedrius Alkauskas, Congruence Properties of the Function that Counts Compositions into Powers of 2 , J. Int. Seq. 13 (2010), 10.5.3.
Joerg Arndt, Matters Computational (The Fxtbook), section 38.1, p.729.
Scott M. Bailey and Donald M. Larson, The A(1)-module structure of the homology of Brown-Gitler spectra, arXiv:2107.01316 [math.AT], 2021.
Valentin P. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.
Philippe Biane, Laver tables and combinatorics, arXiv:1810.00548 [math.CO], 2018. Mentions this sequence.
Peter J. Cameron, Firdous Ee Jannat, Rajat Kanti Nath, and Reza Sharafdini, A survey on conjugacy class graphs of groups, arXiv:2403.09423 [math.GR], 2024.
Karl Dilcher and Larry Ericksen, Polynomial Analogues of Restricted b-ary Partition Functions, J. Int. Seq., Vol. 22 (2019), Article 19.3.2.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 48, 581.
Maciej Gawron, Piotr Miska and Maciej Ulas, Arithmetic properties of coefficients of power series expansion of Prod_{n>=0} (1-x^(2^n))^t, arXiv:1703.01955 [math.NT], 2017.
Michael D. Hirschhorn and James A. Sellers, A different view of m-ary partitions, Australasian J. Combin., vol.30 (2004), 193-196.
Jonathan Jordan and Richard Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. doi: 10.4236/am.2010.15045. - From _N. J. A. Sloane_, Feb 03 2013
Yasuyuki Kachi and Pavlos Tzermias, On the m-ary partition numbers, Algebra and Discrete Mathematics, Volume 19 (2015). Number 1, pp. 67-76.
Matjaž Konvalinka and Igor Pak, Cayley compositions, partitions, polytopes, and geometric bijections, Journal of Combinatorial Theory, Series A, Volume 123, Issue 1, April 2014, Pages 86-91; see also DOI link. - From _N. J. A. Sloane_, Dec 22 2012
Apisit Pakapongpun and Thomas Ward, Functorial Orbit counting, J. Int. Seq., 12 (2009) 09.2.4, example 25.
Øystein J. Rodseth and James A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.
David Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (No. 8, 2002), 887-895; see p. 888.
N. J. A. Sloane and James A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003.
N. J. A. Sloane and James A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

A000123 is the main entry for the binary partition function and gives many more properties and references.
Cf. A115625 (labeled binary partitions), A115626 (labeled non-squashing partitions).
Convolution inverse of A106400.
Cf. A023893, A062051, A105420, A131995, A040039, A018819, A088567, A089054, A115361, A168261, A171238, A179051, A008619.
Multiplicity of n in A048675, for distinct prime indices A087207.
Row lengths of A277905.
A118462 lists binary ranks of strict integer partitions, row sums A372888.
A372890 adds up binary ranks of integer partitions.
Binary indices: A000120, A001511, A029931, A048793, A070939, A272020.
Cf. A005940, A019565, A056239, A158704, A158705, A231204, A277319, A372688.

a018819 n = a018819_list !! n
a018819_list = 1 : f (tail a008619_list) where
   f (x:xs) = (sum $ take x a018819_list) : f xs
-- Reinhard Zumkeller, Jan 28 2012

import Data.List (intersperse)
a018819 = (a018819_list !!)
a018819_list = 1 : 1 : (<*>) (zipWith (+)) (intersperse 0) (tail a018819_list)
-- Johan Wiltink, Nov 08 2018

with(combinat); N:=8; a:=array(1..N); c:=array(1..N);
for n from 1 to N do p:=partition(n); np:=nops(p); t:=0;
for s to np do r:=p[s]; r:=sort(r,`>`); nr:=nops(r); j:=1;
# while jsum(r[k],k=j+1..nr) do j:=j+1;od; # gives A040039
while j= sum(r[k],k=j+1..nr) do j:=j+1;od; # gives A018819
if j=nr then t:=t+1;fi od; a[n]:=t; od; # John McKay

max = 59; a[0] = a[1] = 1; a[n_?OddQ] := a[n] = a[n-1]; a[n_?EvenQ] := a[n] = a[n-1] + a[n/2]; Table[a[n], {n, 0, max}]
(* or *) CoefficientList[Series[1/Product[(1-x^(2^j)), {j, 0, Log[2, max] // Ceiling}], {x, 0, max}], x] (* Jean-François Alcover, May 17 2011, updated Feb 17 2014 *)
a[ n_] := If[n<1, Boole[n==0], a[n] = a[n-1] + If[EvenQ@n, a[Quotient[n,2]], 0]]; (* Michael Somos, May 04 2022 *)
Table[Count[IntegerPartitions[n],?(AllTrue[Log2[#],IntegerQ]&)],{n,0,60}] (* _Harvey P. Dale, Jun 20 2024 *)

{ n=15; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]*2)); c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } /* Jon Perry */

{a(n) = my(A, m); if( n<1, n==0, m=1; A = 1 + O(x); while(m<=n, m*=2; A = subst(A, x, x^2) / (1 - x)); polcoeff(A, n))}; /* Michael Somos, Aug 25 2003 */

{a(n) = if( n<1, n==0, if( n%2, a(n-1), a(n/2)+a(n-1)))}; /* Michael Somos, Aug 25 2003 */

from functools import lru_cache
@lru_cache(maxsize=None)
def A018819(n): return 1 if n == 0 else A018819(n-1) + (0 if n % 2 else A018819(n//2)) # Chai Wah Wu, Jan 18 2022

a(2m+1) = a(2m), a(2m) = a(2m-1) + a(m). Proof: If n is odd there is a part of size 1; removing it gives a partition of n - 1. If n is even either there is a part of size 1, whose removal gives a partition of n - 1, or else all parts have even sizes and dividing each part by 2 gives a partition of n/2.
G.f.: 1 / Product_{j>=0} (1-x^(2^j)).
a(n) = (1/n)*Sum_{k = 1..n} A038712(k)*a(n-k), n > 1, a(0) = 1. - Vladeta Jovovic, Aug 22 2002
a(2*n) = a(2*n + 1) = A000123(n). - Michael Somos, Aug 25 2003
a(n) = 1 if n = 0, Sum_{j = 0..floor(n/2)} a(j) if n > 0. - David W. Wilson, Aug 16 2007
G.f. A(x) satisfies A(x^2) = (1-x) * A(x). - Michael Somos, Aug 25 2003
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*w - 2*u*v^2 + v^3. - Michael Somos, Apr 10 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u6 * u1^3 - 3*u3*u2*u1^2 + 3*u3*u2^2*u1 - u3*u2^3. - Michael Somos, Oct 15 2006
G.f.: 1/( Sum_{n >= 0} x^evil(n) - x^odious(n) ), where evil(n) = A001969(n) and odious(n) = A000069(n). - Paul D. Hanna, Jan 23 2012
Let A(x) by the g.f. and B(x) = A(x^k), then 0 = B*((1-A)^k - (-A)^k) + (-A)^k, see fxtbook link. - Joerg Arndt, Dec 17 2012
G.f.: Product_{n>=0} (1+x^(2^n))^(n+1), see the fxtbook link. - Joerg Arndt, Feb 28 2014
G.f.: 1 + Sum_{i>=0} x^(2^i) / Product_{j=0..i} (1 - x^(2^j)). - Ilya Gutkovskiy, May 07 2017

A230877 If n = Sum_{i=0..m} c(i)2^i, c(i) = 0 or 1, then a(n) = Sum_{i=0..m} (m+1-i)c(i).

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 3, 6, 1, 5, 4, 8, 3, 7, 6, 10, 1, 6, 5, 10, 4, 9, 8, 13, 3, 8, 7, 12, 6, 11, 10, 15, 1, 7, 6, 12, 5, 11, 10, 16, 4, 10, 9, 15, 8, 14, 13, 19, 3, 9, 8, 14, 7, 13, 12, 18, 6, 12, 11, 17, 10, 16, 15, 21, 1, 8, 7, 14, 6, 13, 12, 19, 5, 12, 11, 18, 10, 17, 16, 23, 4, 11, 10, 17, 9, 16, 15, 22, 8, 15, 14

Offset: 0

N. J. A. Sloane, Nov 08 2013

Suggested by Jon Perry's A231204, only now the leading power of 2 gets weight 1.

			For n=13 we have 1101, so we add 1+2+4, getting a(13)=7.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A029931, A231204, A329752.

f:=proc(n) local t1,m,i;
t1:=convert(n,base,2);
m:=nops(t1)-1;
add((m+1-i)*t1[i+1], i=0..m);
end;

Array[Total[Position[IntegerDigits[#, 2], 1], 2] &, 100, 0] (* Paolo Xausa, Mar 18 2024 *)

a(n) = { my (b=binary(n)); sum(k=1, #b, b[k]*k) } \\ Rémy Sigrist, Jun 25 2021

def A230877(n): return sum(i for i, j in enumerate(bin(n)[2:],1) if j=='1') # Chai Wah Wu, Jan 09 2023

A372429 Sum of binary indices of prime(n). Sum of positions of ones in the reversed binary expansion of prime(n).

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 6, 8, 11, 13, 15, 10, 11, 13, 16, 15, 18, 19, 10, 13, 12, 17, 15, 17, 14, 17, 19, 20, 21, 19, 28, 11, 13, 15, 17, 19, 21, 17, 20, 22, 22, 23, 29, 16, 19, 21, 23, 30, 24, 25, 26, 31, 27, 33, 10, 15, 17, 19, 18, 19, 21, 19, 23, 26, 25, 28, 23

Offset: 1

Gus Wiseman, May 02 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Do 2, 3, 4, 7, 12, 14 appear just once?
Are 1, 5, 9 missing?
The above questions hold true up to n = 10^6. - John Tyler Rascoe, May 21 2024

			The primes together with their binary expansions and binary indices begin:
   2:      10 ~ {2}
   3:      11 ~ {1,2}
   5:     101 ~ {1,3}
   7:     111 ~ {1,2,3}
  11:    1011 ~ {1,2,4}
  13:    1101 ~ {1,3,4}
  17:   10001 ~ {1,5}
  19:   10011 ~ {1,2,5}
  23:   10111 ~ {1,2,3,5}
  29:   11101 ~ {1,3,4,5}
  31:   11111 ~ {1,2,3,4,5}
  37:  100101 ~ {1,3,6}
  41:  101001 ~ {1,4,6}
  43:  101011 ~ {1,2,4,6}
  47:  101111 ~ {1,2,3,4,6}
  53:  110101 ~ {1,3,5,6}
  59:  111011 ~ {1,2,4,5,6}
  61:  111101 ~ {1,3,4,5,6}
  67: 1000011 ~ {1,2,7}
  71: 1000111 ~ {1,2,3,7}
  73: 1001001 ~ {1,4,7}
  79: 1001111 ~ {1,2,3,4,7}

John Tyler Rascoe, Table of n, a(n) for n = 1..9438

The number instead of sum of binary indices is A014499.
Restriction of A029931 (sum of binary indices) to the primes A000040.
The maximum instead of sum of binary indices is A035100, see also A023506.
Row-sums of A372471.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020.
A056239 adds up prime indices.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
A372427 lists numbers whose binary and prime indices have the same sum.
Cf. A005940, A030101, A059893, A071814, A230877, A231204, A358134, A359359, A359401, A372430-A372437, A372441.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Total[bix[Prime[n]]],{n,100}]

a(n) = A029931(prime(n)).

A359678 Number of multisets (finite weakly increasing sequences of positive integers) with zero-based weighted sum (A359674) equal to n > 0.

Original entry on oeis.org

1, 2, 4, 4, 6, 9, 8, 10, 14, 13, 16, 21, 17, 22, 28, 23, 30, 37, 30, 38, 46, 38, 46, 59, 46, 55, 70, 59, 70, 86, 67, 81, 96, 84, 98, 115, 95, 114, 135, 114, 132, 158, 127, 156, 178, 148, 176, 207, 172, 201, 227, 196, 228, 270, 222, 255, 296, 255, 295, 338, 278

Offset: 1

Gus Wiseman, Jan 15 2023

nonn

The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

			The a(1) = 1 through a(8) = 10 multisets:
  {1,1}  {1,2}  {1,3}    {1,4}  {1,5}    {1,6}      {1,7}    {1,8}
         {2,2}  {2,3}    {2,4}  {2,5}    {2,6}      {2,7}    {2,8}
                {3,3}    {3,4}  {3,5}    {3,6}      {3,7}    {3,8}
                {1,1,1}  {4,4}  {4,5}    {4,6}      {4,7}    {4,8}
                                {5,5}    {5,6}      {5,7}    {5,8}
                                {1,1,2}  {6,6}      {6,7}    {6,8}
                                         {1,2,2}    {7,7}    {7,8}
                                         {2,2,2}    {1,1,3}  {8,8}
                                         {1,1,1,1}           {1,2,3}
                                                             {2,2,3}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

The one-based version is A320387.
Number of appearances of n > 0 in A359674.
The sorted minimal ranks are A359675, reverse A359680.
The minimal ranks are A359676, reverse A359681.
The maximal ranks are A359757.
A053632 counts compositions by zero-based weighted sum.
A124757 gives zero-based weighted sums of standard compositions, rev A231204.
Cf. A029931, A243055, A304818, A358194, A359677.

zz[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&&GreaterEqual @@ Differences[Append[#,0]]&];
Table[Sum[Append[z,0][[1]]-Append[z,0][[2]],{z,zz[n]}],{n,30}]

seq(n)={Vec(sum(k=2, (sqrtint(8*n+1)+1)\2, my(t=binomial(k, 2)); x^t/((1-x^t)*prod(j=1, k-1, 1 - x^(t-binomial(j, 2)) + O(x^(n-t+1))))))} \\ Andrew Howroyd, Jan 22 2023

G.f.: Sum_{k>=2} x^binomial(k,2)/((1 - x^binomial(k,2))*Product_{j=1..k-1} (1 - x^(binomial(k,2)-binomial(j,2)))). - Andrew Howroyd, Jan 22 2023

A264034 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A161680(n)) is the number of integer partitions of n with weighted sum k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 3, 2, 1

Offset: 0

Christian Stump, Nov 01 2015

Row sums give A000041.
The weighted sum is given by the sum of the rows where row i is weighted by i.
Note that the first part has weight 0. This statistic (zero-based weighted sum) is ranked by A359677, reverse A359674. Also the number of partitions of n with one-based weighted sum n + k. - Gus Wiseman, Jan 10 2023

			Triangle T(n,k) begins:
  1;
  1;
  1,1;
  1,1,0,1;
  1,1,1,1,0,0,1;
  1,1,1,1,1,0,1,0,0,0,1;
  1,1,1,2,1,0,2,1,0,0,1,0,0,0,0,1;
  1,1,1,2,1,1,2,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1;
  1,1,1,2,2,1,2,2,1,1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0,0,0,0,1;
  ...
The a(15,31) = 5 partitions of 15 with weighted sum 31 are: (6,2,2,1,1,1,1,1), (5,4,1,1,1,1,1,1), (5,2,2,2,2,1,1), (4,3,2,2,2,2), (3,3,3,3,2,1). These are also the partitions of 15 with one-based weighted sum 46. - _Gus Wiseman_, Jan 09 2023

Alois P. Heinz, Rows n = 0..50, flattened
FindStat - Combinatorial Statistic Finder, Weighted size of a partition

Row sums are A000041.
The version for compositions is A053632, ranked by A124757 (reverse A231204).
Row lengths are A152947, or A161680 plus 1.
The one-based version is also A264034, if we use k = n..n(n+1)/2.
The reverse version A358194 counts partitions by sum of partial sums.
A359677 gives zero-based weighted sum of prime indices, reverse A359674.
A359678 counts multisets by zero-based weighted sum.
Cf. A029931, A066185, A124757, A304818, A318283, A337206, A359042.

b:= proc(n, i, w) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, w)+
      `if`(i>n, 0, x^(w*i)*b(n-i, i, w+1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 01 2015

b[n_, i_, w_] := b[n, i, w] = Expand[If[n == 0, 1, If[i < 1, 0, b[n, i - 1, w] + If[i > n, 0, x^(w*i)*b[n - i, i, w + 1]]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Total[Accumulate[Reverse[#]]]==k&]],{n,0,8},{k,n,n*(n+1)/2}] (* Gus Wiseman, Jan 09 2023 *)

From Alois P. Heinz, Jan 20 2023: (Start)
max_{k=0..A161680(n)} T(n,k) = A337206(n).
Sum_{k=0..A161680(n)} k * T(n,k) = A066185(n). (End)

A161511 Number of 1...0 pairs in the binary representation of 2n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 6, 4, 5, 5, 6, 5, 7, 6, 7, 5, 8, 7, 8, 6, 9, 7, 8, 5, 6, 6, 7, 6, 8, 7, 8, 6, 9, 8, 9, 7, 10, 8, 9, 6, 10, 9, 10, 8, 11, 9, 10, 7, 12, 10, 11, 8, 12, 9, 10, 6, 7, 7, 8, 7, 9, 8, 9, 7, 10, 9, 10, 8, 11, 9, 10, 7, 11, 10, 11, 9, 12, 10, 11, 8, 13, 11, 12, 9, 13

Offset: 0

Franklin T. Adams-Watters, Jun 11 2009

Row (partition) sums of A125106.

a(n) is also the weight (= sum of parts) of the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017

			For n = 5, the binary representation of 2n is 1010; the 1...0 pairs are 10xx, 1xx0, and xx10, so a(5) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Peter J. Taylor, Recursion for A161511, answer to question on MathOverflow (2025).

Cf. A000120, A243499 (gives the corresponding products), A227183, A056239, A243503, A006068, A163511.
Sum of prime indices of A005940.
Row sums of A125106.
A reverse version is A359043, row sums of A242628.
A029837 adds up standard compositions, row sums of A066099.
A029931 adds up binary indices, row sums of A048793.
Cf. A059893, A228351, A230877, A231204, A253565, A358194, A359042, A359359.

a[0] = 0; a[n_] := If[EvenQ[n], a[n/2] + DigitCount[n/2, 2, 1], a[(n-1)/2] + 1]; Array[a, 93, 0] (* Jean-François Alcover, Sep 09 2017 *)

a(n)=local(t,k);t=0;k=1;while(n>0,if(n%2==0,k++,t+=k);n\=2);t

def A161511(n):
    a, b = 0, 0
    for i, j in enumerate(bin(n)[:1:-1], 1):
        if int(j):
            a += i-b
            b += 1
    return a # Chai Wah Wu, Jul 26 2023

;; Two variants, the recursive one requiring memoizing definec-macro from Antti Karttunen's IntSeq-library.
(define (A161511 n) (let loop ((n n) (i 1) (s 0)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) (+ i 1) s)) (else (loop (/ (- n 1) 2) i (+ s i))))))
(definec (A161511 n) (cond ((zero? n) n) ((even? n) (+ (A000120 n) (A161511 (/ n 2)))) (else (+ 1 (A161511 (/ (- n 1) 2))))))
;; Antti Karttunen, Jun 28 2014

a(0) = 0; a(2n) = a(n) + A000120(n); a(2n+1) = a(n) + 1.
From Antti Karttunen, Jun 28 2014: (Start)
Can be also obtained by mapping with an appropriate permutation from the lists of partition sizes computed for other enumerations similar to A125106:
a(n) = A227183(A006068(n)).
a(n) = A056239(A005940(n+1)).
a(n) = A243503(A163511(n)). (End)
a(n) = A029931(n) - binomial(A000120(n),2). - Gus Wiseman, Jan 03 2023
a(n) = a(n - A048896(n-1)) + 1 for n>=1 (see Peter J. Taylor link). - Mikhail Kurkov, Jul 04 2025

A372471 Irregular triangle read by rows where row n lists the binary indices of the n-th prime number.

Original entry on oeis.org

2, 1, 2, 1, 3, 1, 2, 3, 1, 2, 4, 1, 3, 4, 1, 5, 1, 2, 5, 1, 2, 3, 5, 1, 3, 4, 5, 1, 2, 3, 4, 5, 1, 3, 6, 1, 4, 6, 1, 2, 4, 6, 1, 2, 3, 4, 6, 1, 3, 5, 6, 1, 2, 4, 5, 6, 1, 3, 4, 5, 6, 1, 2, 7, 1, 2, 3, 7, 1, 4, 7, 1, 2, 3, 4, 7, 1, 2, 5, 7, 1, 4, 5, 7, 1, 6, 7

Offset: 1

Gus Wiseman, May 07 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			We have prime(12) = (2^1 + 2^3 + 2^6)/2, so row 12 is (1,3,6).
Each prime followed by its binary indices:
   2: 2
   3: 1 2
   5: 1 3
   7: 1 2 3
  11: 1 2 4
  13: 1 3 4
  17: 1 5
  19: 1 2 5
  23: 1 2 3 5
  29: 1 3 4 5
  31: 1 2 3 4 5
  37: 1 3 6
  41: 1 4 6
  43: 1 2 4 6
  47: 1 2 3 4 6

Row lengths are A014499.
Second column is A023506(n) + 1.
Final column is A035100.
Prime-indexed rows of A048793.
Row-sums are A372429, restriction of A029931 (sum of binary indices).
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020.
A070939 gives length of binary expansion.
Cf. A000040, A005940, A056239, A071814, A096111, A191232, A230877, A231204, A372427-A372442.

Table[Join@@Position[Reverse[IntegerDigits[Prime[n],2]],1],{n,15}]

A359674 Zero-based weighted sum of the prime indices of n in weakly increasing order.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 2, 3, 0, 5, 0, 4, 3, 6, 0, 6, 0, 7, 4, 5, 0, 9, 3, 6, 6, 9, 0, 8, 0, 10, 5, 7, 4, 11, 0, 8, 6, 12, 0, 10, 0, 11, 8, 9, 0, 14, 4, 9, 7, 13, 0, 12, 5, 15, 8, 10, 0, 14, 0, 11, 10, 15, 6, 12, 0, 15, 9, 11, 0, 17, 0, 12, 9, 17, 5, 14, 0, 18

Offset: 1

Gus Wiseman, Jan 13 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

			The prime indices of 12 are {1,1,2}, so a(12) = 0*1 + 1*1 + 2*2 = 5.

Positions of last appearances (except 0) are A001248.
Positions of 0's are A008578.
The version for standard compositions is A124757, reverse A231204.
The one-based version is A304818, reverse A318283.
Positions of first appearances are A359675, reverse A359680.
First position of n is A359676(n), reverse A359681.
The reverse version is A359677, firsts A359679.
Number of appearances of positive n is A359678(n).
A053632 counts compositions by zero-based weighted sum.
A112798 lists prime indices, length A001222, sum A056239.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A029931, A055932, A243055, A359043, A358194, A359360, A359497, A359681, A359682, A359755.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
Table[wts[primeMS[n]],{n,100}]

A359402 Numbers whose binary expansion and reversed binary expansion have the same sum of positions of 1's, where positions in a sequence are read starting with 1 from the left.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, 45, 51, 63, 65, 70, 73, 78, 85, 93, 99, 107, 119, 127, 129, 150, 153, 165, 189, 195, 219, 231, 255, 257, 266, 273, 282, 294, 297, 310, 313, 325, 334, 341, 350, 355, 365, 371, 381, 387, 397, 403, 413, 427, 443, 455, 471

Offset: 1

Gus Wiseman, Jan 05 2023

nonn

Also numbers whose binary expansion and reversed binary expansion have the same sum of partial sums.
Also numbers whose average position of a 1 in their binary expansion is (c+1)/2, where c is the number of digits.
Conjecture: Also numbers whose binary expansion has as least squares fit a line of zero slope, counted by A222955.

			The binary expansion of 70 is (1,0,0,0,1,1,0), with positions of 1's {1,5,6}, while the reverse positions are {2,3,7}. Both sum to 12, so 70 is in the sequence.

Binary words of this type appear to be counted by A222955.
For greater instead of equal sums we have A359401.
These are the indices of 0's in A359495.
A030190 gives binary expansion, reverse A030308.
A048793 lists partial sums of reversed standard compositions, sums A029931.
A070939 counts binary digits, 1's A000120.
A326669 lists numbers with integer mean position of a 1 in binary expansion.
Cf. A051293, A053632, A231204, A291166, A304818, A318283, A326672, A326673, A358134, A359042.

Select[Range[0,100],#==0||Mean[Join@@Position[IntegerDigits[#,2],1]]==(IntegerLength[#,2]+1)/2&]

from functools import reduce
from itertools import count, islice
def A359402_gen(startvalue=0): # generator of terms
    return filter(lambda n:(r:=reduce(lambda c, d:(c[0]+d[0]*(e:=int(d[1])),c[1]+e),enumerate(bin(n)[2:],start=1),(0,0)))[0]<<1==(n.bit_length()+1)*r[1],count(max(startvalue,0)))
A359402_list = list(islice(A359402_gen(),30)) # Chai Wah Wu, Jan 08 2023

A230877(a(n)) = A029931(a(n)).

A359676 Least positive integer whose weakly increasing prime indices have zero-based weighted sum n (A359674).

Original entry on oeis.org

1, 4, 6, 8, 14, 12, 16, 20, 30, 24, 32, 36, 40, 52, 48, 56, 100, 72, 80, 92, 96, 104, 112, 124, 136, 148, 176, 152, 214, 172, 184, 188, 262, 212, 272, 236, 248, 244, 304, 268, 346, 284, 328, 292, 386, 316, 398, 332, 376, 356, 458, 388, 478, 404, 472, 412, 526

Offset: 1

Gus Wiseman, Jan 14 2023

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

			The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   14: {1,4}
   12: {1,1,2}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   52: {1,1,6}
   48: {1,1,1,1,2}

First position of n in A359674, reverse A359677.
The sorted version is A359675, reverse A359680.
The reverse one-based version is A359679, sorted A359754.
The reverse version is A359681.
The one-based version is A359682, sorted A359755.
The version for standard compositions is A359756, one-based A089633.
A053632 counts compositions by zero-based weighted sum.
A112798 lists prime indices, length A001222, sum A056239.
A124757 gives zero-based weighted sum of standard compositions, rev A231204.
A304818 gives weighted sums of prime indices, reverse A318283.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A001248, A029931, A055932, A243055, A359043, A358194, A359497, A359683.

nn=20;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
seq=Table[wts[primeMS[n]],{n,1,Prime[nn]^2}];
Table[Position[seq,k][[1,1]],{k,0,nn}]

cp's OEIS Frontend

A018819 Binary partition function: number of partitions of n into powers of 2.

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A230877 If n = Sum_{i=0..m} c(i)*2^i, c(i) = 0 or 1, then a(n) = Sum_{i=0..m} (m+1-i)*c(i).

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A372429 Sum of binary indices of prime(n). Sum of positions of ones in the reversed binary expansion of prime(n).

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A359678 Number of multisets (finite weakly increasing sequences of positive integers) with zero-based weighted sum (A359674) equal to n > 0.

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A264034 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A161680(n)) is the number of integer partitions of n with weighted sum k.

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A161511 Number of 1...0 pairs in the binary representation of 2n.

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A230877 If n = Sum_{i=0..m} c(i)2^i, c(i) = 0 or 1, then a(n) = Sum_{i=0..m} (m+1-i)c(i).