A029931 -id:A029931 - cp's OEIS frontend

A233249 a(1)=0; for k >= 1, let prime(k) map to 10...0 with k-1 zeros and let prime(k)*prime(m) map to the concatenation in binary of 2^(k-1) and 2^(m-1). For n >= 2, let the prime power factorization of n be mapped to r(n). a(n) is the term in A114994 which is c-equivalent to r(n) (see there our comment).

0, 1, 2, 3, 4, 5, 8, 7, 10, 9, 16, 11, 32, 17, 18, 15, 64, 21, 128, 19, 34, 33, 256, 23, 36, 65, 42, 35, 512, 37, 1024, 31, 66, 129, 68, 43, 2048, 257, 130, 39, 4096, 69, 8192, 67, 74, 513, 16384, 47, 136, 73, 258, 131, 32768, 85, 132, 71, 514, 1025, 65536, 75

Offset: 1

Vladimir Shevelev, Dec 06 2013

Let (10...0)_i (i>=0) denote 2^i in binary. Under (10...0)_i^k we understand a concatenation of (10...0)_i k times.
If n=Product_{i=1..m} p_i^t_i is the prime power factorization of n, then in the name r(n)=concatenation{i=1..m} ((10...0_(i-1)^t_i).
Numbers q and s are called c-equivalent if their binary expansions contain the same set of parts of the form 10...0. For example, 14=(1)(1)(10)~(10)(1)(1)=11.
Conversely, if n~n_1 such that n_1 is in A114994 and has c-factorization: n_1 = concatenation{i=m,...,0} ((10...0)i^t_i), one can consider "converse" sequence {s(n)}, where s(n) = Product{i=m..0} p_(i+1)^t_i.
For example, for n=22, n_1=21=((10)^2)(1), and s(22)=3^2*2=18.
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary binary expansion of k, prepending 0, taking first differences, and reversing again. Then a(n) is the number k such that the k-th composition in standard order consists of the prime indices of n in weakly decreasing order (the partition with Heinz number n). - Gus Wiseman, Apr 02 2020

			n=10=2*5 is mapped to (1)(100)~(100)(1). Since 9 is in A114994, then a(10)=9.
From _Gus Wiseman_, Apr 02 2020: (Start)
The sequence together with the corresponding compositions begins:
   0: ()             128: (8)             2048: (12)
   1: (1)             19: (3,1,1)          257: (8,1)
   2: (2)             34: (4,2)            130: (6,2)
   3: (1,1)           33: (5,1)             39: (3,1,1,1)
   4: (3)            256: (9)             4096: (13)
   5: (2,1)           23: (2,1,1,1)         69: (4,2,1)
   8: (4)             36: (3,3)           8192: (14)
   7: (1,1,1)         65: (6,1)             67: (5,1,1)
  10: (2,2)           42: (2,2,2)           74: (3,2,2)
   9: (3,1)           35: (4,1,1)          513: (9,1)
  16: (5)            512: (10)           16384: (15)
  11: (2,1,1)         37: (3,2,1)           47: (2,1,1,1,1)
  32: (6)           1024: (11)             136: (4,4)
  17: (4,1)           31: (1,1,1,1,1)       73: (3,3,1)
  18: (3,2)           66: (5,2)            258: (7,2)
  15: (1,1,1,1)      129: (7,1)            131: (6,1,1)
  64: (7)             68: (4,3)          32768: (16)
  21: (2,2,1)         43: (2,2,1,1)         85: (2,2,2,1)
For example, the Heinz number of (2,2,1) is 18, and the 21st composition in standard order is (2,2,1), so a(18) = 21.
(End)

Peter J. C. Moses, Table of n, a(n) for n = 1..2500

The sorted version is A114994.
The primorials A002110 map to A246534.
A partial inverse is A333219.
The reversed version is A333220.
Cf. A000120, A029931, A035327, A048793, A066099, A070939, A124767, A124768, A228351, A272020, A333217, A333221.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[2^Accumulate[primeMS[n]]]/2,{n,100}] (* Gus Wiseman, Apr 02 2020 *)

A059893(a(n)) = A333220(n). A124767(a(n)) = A001221(n). - Gus Wiseman, Apr 02 2020

More terms from Peter J. C. Moses, Dec 07 2013

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

A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

1, 2, 4, 7, 14, 24, 39, 61, 122, 203, 315, 469, 676, 952, 1307, 1771, 3542, 5708, 8432, 11877, 16123, 21415, 27835, 35757, 45343, 57010, 70778, 87384, 106479, 129304, 155802, 187223, 374446, 588130, 835800, 1124981, 1456282, 1841361, 2281772, 2791896, 3367162

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

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.

			The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {1,3,4}
                         {2,3,4}

Wikipedia, Axiom of choice.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Simple graphs of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A134964, complement A140637.
Simple graphs not of this type are counted by A367867, covering A367868.
Set systems of this type are counted by A367902, ranks A367906.
Set systems not of this type are counted by A367903, ranks A367907.
Set systems uniquely of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368098, complement A368097.
A version for MM-numbers of multisets is A368100, complement A355529.
Factorizations are counted by A368414/A370814, complement A368413/A370813.
For prime indices we have A370582, differences A370586.
The complement for prime indices is A370583, differences A370587.
The complement is A370637, differences A370589, without ones A370643.
The case of a unique choice is A370638, maxima A370640, differences A370641.
First differences are A370639.
The minimal case of the complement is A370642, without ones A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368095, A368109.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]

a(2^n - 1) = A367902(n).

Partial sums of A370639.

a(19)-a(40) from Alois P. Heinz, Mar 09 2024

A231204 If n = Sum_{i=0..m} c(i)2^i, c(i) = 0 or 1, then a(n) = Sum_{i=0..m} (m-i)c(i).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 3, 0, 3, 2, 5, 1, 4, 3, 6, 0, 4, 3, 7, 2, 6, 5, 9, 1, 5, 4, 8, 3, 7, 6, 10, 0, 5, 4, 9, 3, 8, 7, 12, 2, 7, 6, 11, 5, 10, 9, 14, 1, 6, 5, 10, 4, 9, 8, 13, 3, 8, 7, 12, 6, 11, 10, 15, 0, 6, 5, 11, 4, 10, 9, 15, 3, 9, 8, 14, 7, 13, 12, 18, 2, 8, 7, 13, 6, 12, 11, 17, 5, 11, 10, 16, 9, 15, 14, 20, 1, 7, 6, 12

Offset: 0

Jon Perry, Nov 05 2013

A literal interpretation of the binary numbers.

Sum of the number of digits to the left (exclusive) of each 1 in the binary expansion of n. - Gus Wiseman, Jan 09 2023

			For n=13 we have 1101, so we add 0+1+3=4, getting a(13)=4.

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

Cf. A000120, A029931, A230877, A384868

Cf. A000523.

for (i=0;i<100;i++) {
s=i.toString(2);
o=0;
sl=s.length;
for (j=0;j

f:=proc(n) local t1,m,i;
t1:=convert(n,base,2);
m:=nops(t1)-1;
add((m-i)*t1[i+1], i=0..m);
end; # N. J. A. Sloane, Nov 08 2013

Table[Total[Join@@Position[IntegerDigits[n,2],1]-1],{n,0,100}] (* Gus Wiseman, Jan 09 2023 *)

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

def A230204(n): return sum(i for i, j in enumerate(bin(n)[2:]) if j=='1') # Chai Wah Wu, Jan 09 2023

a(n) = A230877(n) - A000120(n). - Gus Wiseman, Jan 09 2023

Edited by N. J. A. Sloane, Nov 08 2013

Name edited by Gus Wiseman, Jan 09 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)

A333223 Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 24, 26, 28, 31, 32, 33, 34, 35, 36, 40, 41, 42, 48, 50, 56, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 84, 85, 88, 96, 98, 100, 104, 106, 112, 120, 127, 128, 129, 130, 131, 132, 133

Offset: 1

Gus Wiseman, Mar 17 2020

nonn

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

			The list of terms together with the corresponding compositions begins:
    0: ()            21: (2,2,1)           65: (6,1)
    1: (1)           24: (1,4)             66: (5,2)
    2: (2)           26: (1,2,2)           67: (5,1,1)
    3: (1,1)         28: (1,1,3)           68: (4,3)
    4: (3)           31: (1,1,1,1,1)       69: (4,2,1)
    5: (2,1)         32: (6)               70: (4,1,2)
    6: (1,2)         33: (5,1)             71: (4,1,1,1)
    7: (1,1,1)       34: (4,2)             72: (3,4)
    8: (4)           35: (4,1,1)           73: (3,3,1)
    9: (3,1)         36: (3,3)             74: (3,2,2)
   10: (2,2)         40: (2,4)             80: (2,5)
   12: (1,3)         41: (2,3,1)           81: (2,4,1)
   15: (1,1,1,1)     42: (2,2,2)           84: (2,2,3)
   16: (5)           48: (1,5)             85: (2,2,2,1)
   17: (4,1)         50: (1,3,2)           88: (2,1,4)
   18: (3,2)         56: (1,1,4)           96: (1,6)
   19: (3,1,1)       63: (1,1,1,1,1,1)     98: (1,4,2)
   20: (2,3)         64: (7)              100: (1,3,3)

Distinct subsequences are counted by A124770 and A124771.
A superset of A333222, counted by A169942, with partition case A325768.
These compositions are counted by A325676.
A version for partitions is A325769, with Heinz numbers A325778.
The number of distinct positive subsequence-sums is A333224.
The number of distinct subsequence-sums is A333257.
Numbers whose binary indices are a strict knapsack partition are A059519.
Knapsack partitions are counted by A108917, with strict case A275972.
Golomb subsets are counted by A143823.
Heinz numbers of knapsack partitions are A299702.
Maximal Golomb rulers are counted by A325683.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295 A325779, A233564, A325680, A325687, A325770, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Total/@Union[ReplaceList[stc[#],{_,s__,_}:>{s}]]&]

A164894 Base-10 representation of the binary string formed by appending 10, 100, 1000, 10000, ..., etc., to 1.

Original entry on oeis.org

1, 6, 52, 840, 26896, 1721376, 220336192, 56406065280, 28879905423616, 29573023153783296, 60565551418948191232, 248076498612011791288320, 2032242676629600594233921536, 33296264013899376135928570454016, 1091051979207454757222107396637212672

Offset: 1

Gil Broussard, Aug 29 2009

These numbers are half the sum of powers of 2 indexed by differences of a triangular number and each smaller triangular number (e.g., 21 - 15 = 6, 21 - 10 = 11, ..., 21 - 0 = 21).
This suggests another way to think about these numbers: consider the number triangle formed by the characteristic function of the triangular numbers (A010054), join together the first n rows (the very first row is row 0) as a single binary string and that gives the (n + 1)th term of this sequence. - Alonso del Arte, Nov 15 2013
Numbers k such that the k-th composition in standard order (row k of A066099) is an initial interval. - Gus Wiseman, Apr 02 2020

			a(1) = 1, also 1 in binary.
a(2) = 6, or 110 in binary.
a(3) = 52, or 110100 in binary.
a(4) = 840, or 1101001000 in binary.

Harvey P. Dale, Table of n, a(n) for n = 1..81

The version for prime (rather than binary) indices is A002110.
The non-strict generalization is A225620.
The reversed version is A246534.
Standard composition numbers of permutations are A333218.
Standard composition numbers of strict increasing compositions are A333255.
Cf. A000120, A029931, A048793, A066099, A070939, A124768, A233564, A272919, A333217, A333220, A333379.

Table[Sum[2^((n^2 + n)/2 - (k^2 + k)/2 - 1), {k, 0, n - 1}], {n, 25}] (* Alonso del Arte, Nov 14 2013 *)
Module[{nn=15,t},t=Table[10^n,{n,0,nn}];Table[FromDigits[Flatten[IntegerDigits/@Take[t,k]],2],{k,nn}]] (* Harvey P. Dale, Jan 16 2024 *)

def a(n): return int("".join("1"+"0"*i for i in range(n)), 2)
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Jul 05 2021

def A164894(n): return sum(1<<(k*((n<<1)-k-1)>>1)+n-1 for k in range(n)) # Chai Wah Wu, Jul 11 2025

a(n) = Sum_{k=0..n-1} 2^((n^2 + n)/2 - (k^2 + k)/2 - 1). - Alonso del Arte, Nov 15 2013
Intersection of A333255 and A333217. - Gus Wiseman, Apr 02 2020
a(n) = Sum_{k=0..n-1} 2^(k*(2*n-k-1)/2+n-1). - Chai Wah Wu, Jul 11 2025

A372433 Binary weight (number of ones in binary expansion) of the n-th squarefree number.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 04 2024

Harvey P. Dale, Table of n, a(n) for n = 1..1000
MathOverflow, Are there primes of every Hamming weight?
Wikipedia, Hamming weight.

Restriction of A000120 to A005117.
For prime instead of squarefree we have A014499, zeros A035103.
Counting zeros instead of ones gives A372472, cf. A023416, A372473.
For binary length instead of weight we have A372475.
A003714 lists numbers with no successive binary indices.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A145037 counts ones minus zeros in binary expansion, cf. A031443, A031444, A031448, A097110.
A371571 lists positions of zeros in binary expansion, sum A359359.
A371572 lists positions of ones in binary expansion, sum A230877.
A372515 lists positions of zeros in reversed binary expansion, sum A359400.
A372516 counts ones minus zeros in binary expansion of primes, cf. A177718, A177796, A372538, A372539.
Cf. A039004, A049093, A049094, A059015, A069010, A070939, A073642, A211997, A368494, A372474.

DigitCount[Select[Range[100],SquareFreeQ],2,1]
Total[IntegerDigits[#,2]]&/@Select[Range[200],SquareFreeQ] (* Harvey P. Dale, Feb 14 2025 *)

from math import isqrt
from sympy import mobius
def A372433(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m).bit_count() # Chai Wah Wu, Aug 02 2024

a(n) = A000120(A005117(n)).

a(n) + A372472(n) = A372475(n) = A070939(A005117(n)).

A246534 a(n) = Sum_{k=1..n} 2^(T(k)-1), where T(k)=k(k+1)/2 = A000217(k).

Original entry on oeis.org

0, 1, 5, 37, 549, 16933, 1065509, 135283237, 34495021605, 17626681066021, 18032025190548005, 36911520172609651237, 151152638972001256489509, 1238091191924352276155613733, 20283647694843594776223406899749, 664634281540152780046679753547072037

Offset: 0

M. F. Hasler, Aug 28 2014

nonn

Similar to A181388, this occurs as binary encoding of a straight n-omino lying on the y-axis, when the grid points of the first quadrant (N x N, N={0,1,2,...}) are given the weight 2^k, with k=0, 1,2, 3,4,5, ... filled in by antidiagonals.

Numbers k such that the k-th composition in standard order (row k of A066099) is a reversed initial interval. - Gus Wiseman, Apr 02 2020

			Label the cells of an infinite square matrix with 0,1,2,3,... along antidiagonals:
  0 1 3 6 10 ...
  2 4 7 ...
  5 8 ...
  9 ...
  ....
Now any subset of these cells can be represented by the sum of 2 raised to the power written in the given cells. In particular, the subset consisting of the first cell in the first 1, 2, 3, ... rows is represented by 2^0, 2^0+2^2, 2^0+2^2+2^5, ...

The version for prime (rather than binary) indices is A002110.
The non-strict generalization is A114994.
The non-reversed version is A164894.
Intersection of A333256 and A333217.
Partial sums of A036442.
Cf. A000120, A029931, A048793, A066099, A070939, A124766, A228351, A233564, A272919, A333218.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[0,1000],normQ[stc[#]]&&Greater@@stc[#]&] (* Gus Wiseman, Apr 02 2020 *)

t=0;vector(20,n,t+=2^(n*(n+1)/2-1)) \\ yields the vector starting with a[1]=1

t=0;vector(20,n,if(n>1,t+=2^(n*(n-1)/2-1))) \\ yields the vector starting with 0

a = 0
for n in range(1,17): print(a, end =', '); a += 1<<(n-1)*(n+2)//2 # Ya-Ping Lu, Jan 23 2024

A333221 Irregular triangle read by rows where row n lists the set of STC-numbers of permutations of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 10, 9, 12, 16, 11, 13, 14, 32, 17, 24, 18, 20, 15, 64, 21, 22, 26, 128, 19, 25, 28, 34, 40, 33, 48, 256, 23, 27, 29, 30, 36, 65, 96, 42, 35, 49, 56, 512, 37, 38, 41, 44, 50, 52, 1024, 31, 66, 80, 129, 192, 68, 72, 43, 45, 46, 53, 54, 58

Offset: 1

Gus Wiseman, Mar 17 2020

This is a permutation of the nonnegative integers.
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. We define the composition with STC-number k to be the k-th composition in standard order.
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.

			Reading by columns gives:
  0  1  2  3  4  5  8  7  10  9   16  11  32  17  18  15  64  21  128  19
                 6            12      13      24  20          22       25
                                      14                      26       28
  34  33  256  23  36  65  42  35  512  37  1024  31  66  129  68  43
  40  48       27      96      49       38            80  192  72  45
               29              56       41                         46
               30                       44                         53
                                        50                         54
                                        52                         58
The sequence of terms together with the corresponding compositions begins:
     0: ()           24: (1,4)          27: (1,2,1,1)
     1: (1)          18: (3,2)          29: (1,1,2,1)
     2: (2)          20: (2,3)          30: (1,1,1,2)
     3: (1,1)        15: (1,1,1,1)      36: (3,3)
     4: (3)          64: (7)            65: (6,1)
     5: (2,1)        21: (2,2,1)        96: (1,6)
     6: (1,2)        22: (2,1,2)        42: (2,2,2)
     8: (4)          26: (1,2,2)        35: (4,1,1)
     7: (1,1,1)     128: (8)            49: (1,4,1)
    10: (2,2)        19: (3,1,1)        56: (1,1,4)
     9: (3,1)        25: (1,3,1)       512: (10)
    12: (1,3)        28: (1,1,3)        37: (3,2,1)
    16: (5)          34: (4,2)          38: (3,1,2)
    11: (2,1,1)      40: (2,4)          41: (2,3,1)
    13: (1,2,1)      33: (5,1)          44: (2,1,3)
    14: (1,1,2)      48: (1,5)          50: (1,3,2)
    32: (6)         256: (9)            52: (1,2,3)
    17: (4,1)        23: (2,1,1,1)    1024: (11)

Row lengths are A008480.
Column k = 1 is A233249.
Column k = -1 is A333220.
A related triangle for partitions is A215366.
Cf. A000120, A029931, A048793, A056239, A066099, A070939, A112798, A114994, A225620, A228351, A333218, A333219.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fbi[q_]:=If[q=={},0,Total[2^q]/2];
Table[Sort[fbi/@Accumulate/@Permutations[primeMS[n]]],{n,30}]

cp's OEIS Frontend

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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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A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

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A231204 If n = Sum_{i=0..m} c(i)*2^i, c(i) = 0 or 1, then a(n) = Sum_{i=0..m} (m-i)*c(i).

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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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A333223 Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.

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A164894 Base-10 representation of the binary string formed by appending 10, 100, 1000, 10000, ..., etc., to 1.

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A231204 If n = Sum_{i=0..m} c(i)2^i, c(i) = 0 or 1, then a(n) = Sum_{i=0..m} (m-i)c(i).