A038712 -id:A038712 - cp's OEIS frontend

A324056 a(n) = A000593(A005940(1+n)).

1, 1, 4, 1, 6, 4, 13, 1, 8, 6, 24, 4, 31, 13, 40, 1, 12, 8, 32, 6, 48, 24, 78, 4, 57, 31, 124, 13, 156, 40, 121, 1, 14, 12, 48, 8, 72, 32, 104, 6, 96, 48, 192, 24, 248, 78, 240, 4, 133, 57, 228, 31, 342, 124, 403, 13, 400, 156, 624, 40, 781, 121, 364, 1, 18, 14, 56, 12, 84, 48, 156, 8, 112, 72, 288, 32, 372, 104, 320, 6, 168, 96, 384, 48

Offset: 0

Antti Karttunen, Feb 14 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..70327

Cf. A000593, A005940, A038712, A324054.

A000593(n) = sigma(n>>valuation(n, 2)); \\ From A000593
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324056(n) = A000593(A005940(1+n));

a(n) = A000593(A005940(1+n)).

a(n) = A324054(n) / A038712(1+n).

A326988 Sum of nonpowers of 2 dividing n.

Original entry on oeis.org

0, 0, 3, 0, 5, 9, 7, 0, 12, 15, 11, 21, 13, 21, 23, 0, 17, 36, 19, 35, 31, 33, 23, 45, 30, 39, 39, 49, 29, 69, 31, 0, 47, 51, 47, 84, 37, 57, 55, 75, 41, 93, 43, 77, 77, 69, 47, 93, 56, 90, 71, 91, 53, 117, 71, 105, 79, 87, 59, 161, 61, 93, 103, 0, 83, 141, 67, 119, 95, 141, 71, 180, 73, 111, 123, 133, 95, 165, 79, 155

Offset: 1

Omar E. Pol, Aug 18 2019

In other words: a(n) is the sum of the divisors of n that are not powers of 2.
a(n) is also the sum of odd divisors greater than 1 of n, multiplied by the sum of the divisors of n that are powers of 2.
a(n) = 0 if and only if n is a power of 2.
a(n) = n if and only if n is an odd prime.
From Bernard Schott, Sep 17 2019: (Start)
a(n) = 3*n/2 if and only if n is an even semiprime greater than or equal to 6 (A100484).
a(n) = n + sqrt(n) if and only if n is the square of an odd prime (see A001248 without its first term). (End)

			For n = 18 the divisors of 18 are [1, 2, 3, 6, 9, 18]. There are four divisors of 18 that are not powers of 2, they are [3, 6, 9, 18]. The sum of them is 3 + 6 + 9 + 18 = 36, so a(18) = 36.
On the other hand, the sum of odd divisors greater than 1 of 18 is 3 + 9 = 12, and the sum of the divisors of 18 that are powers of 2 is 1 + 2 = 3, then we have that 12 * 3 = 36, so a(18) = 36.

Robert Israel, Table of n, a(n) for n = 1..10000

Row sums of A326989.

Cf. A000079, A000203, A000593, A038712, A057716, A065091, A326987 (number), A326990.

sol:=[];  m:=1;  for n in [1..80] do v:=Set(Divisors(n)) diff {2^k:k in [0..Floor(Log(2,n))]};  sol[m]:=&+v; m:=m+1; end for; sol; // Marius A. Burtea, Aug 24 2019

f:= n -> numtheory:-sigma(n) - 2^(1+padic:-ordp(n,2))+1:
map(f, [$1..100]); # Robert Israel, Apr 29 2020

Table[DivisorSigma[1, n] - Denominator[DivisorSigma[1, 2n]/DivisorSigma[1, n]], {n, 100}] (* Wesley Ivan Hurt, Aug 24 2019 *)

ispp2(n) = (n==1) || (isprimepower(n, &p) && (p==2));
a(n) = sumdiv(n, d, if (!ispp2(d), d)); \\ Michel Marcus, Aug 26 2019

from sympy import divisor_sigma
def A326988(n): return divisor_sigma(n)-(n^(n-1)) # Chai Wah Wu, Aug 04 2022

def divisors(n: Int): IndexedSeq[Int] = (1 to n).filter(n % _ == 0)
(1 to 80).map(divisors().filter(n => n != Integer.highestOneBit(n)).sum) // _Alonso del Arte, Apr 29 2020

a(n) = A000203(n) - A038712(n).
a(n) = (A000593(n) - 1)*A038712(n).
a(n) = A326990(n)*A038712(n).
a(n) = Sum_{d|n, d > 1} d * (1 - [rad(d) = 2]), where rad is the squarefree kernel (A007947) and [] is the Iverson bracket, which gives 1 if the condition is true, 0 if it's false. - Wesley Ivan Hurt, Apr 29 2020

A355584 a(n) is the sum of the 5-smooth divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 1, 15, 13, 18, 1, 28, 1, 3, 24, 31, 1, 39, 1, 42, 4, 3, 1, 60, 31, 3, 40, 7, 1, 72, 1, 63, 4, 3, 6, 91, 1, 3, 4, 90, 1, 12, 1, 7, 78, 3, 1, 124, 1, 93, 4, 7, 1, 120, 6, 15, 4, 3, 1, 168, 1, 3, 13, 127, 6, 12, 1, 7, 4, 18, 1, 195, 1, 3, 124, 7

Offset: 1

Amiram Eldar, Jul 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Sum of the p-smooth divisors of n: A038712 (2), A072079 (3), this sequence (5).

Cf. A000203, A007814, A007949, A051037, A112765, A355582, A355583.

a[n_] := (Times @@ ({2, 3, 5}^(IntegerExponent[n, {2, 3, 5}] + 1) - 1))/8; Array[a, 100]

a(n) = (2^(valuation(n, 2) + 1) - 1) * (3^(valuation(n, 3) + 1) - 1) * (5^(valuation(n, 5) + 1) - 1) / 8;

from sympy import multiplicity as v
def a(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1) * (5**(v(5, n)+1)-1) // 8
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 5, and 1 otherwise.
a(n) = (2^(A007814(n)+1)-1)*(3^(A007949(n)+1)-1)*(5^(A112765(n)+1)-1)/8.
a(n) = A000203(A355582(n)).
a(n) <= A000203(n), with equality if and only if n is in A051037.
Dirichlet g.f.: zeta(s)*(2^s/(2^s-2))*(3^s/(3^s-3))*(5^s/(5^s-5)). - Amiram Eldar, Dec 25 2022

A088841 Numerator of the quotient sigma(7*n)/sigma(n).

Original entry on oeis.org

8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 400, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8, 57, 8, 8, 8, 8, 8, 8

Offset: 1

Labos Elemer, Nov 04 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A088837, A088838, A088839, A088840, A088841, A088842 (denominators).

Cf. A000203, A038712, A080278, A214411, A268354, A283078.

Table[Numerator[DivisorSigma[1, 7*n]/DivisorSigma[1, n]], {n, 1, 128}]

a(n) = numerator(sigma(7*n)/sigma(n)); \\ Amiram Eldar, Mar 22 2024

From Amiram Eldar, Mar 22 2024: (Start)
a(n) = numerator(A283078(n)/A000203(n)).
a(n) = (7^(A214411(n)+2)-1)/6 = (49*A268354(n)-1)/6.
Sum_{k=1..n} a(k) ~ (7/log(7))*n*log(n) + (9/2 + 7*(gamma-1)/log(7))*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A088842(k) = 1 + 36 * Sum_{k>=1} 1/(7^k-1) = 7.87276224676... . (End)

A089312 Write n in binary; a(n) = number represented by rightmost block of 1's.

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 3, 1, 7, 15, 1, 1, 1, 3, 1, 1, 3, 7, 3, 1, 1, 3, 7, 1, 15, 31, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 3, 1, 7, 15, 3, 1, 1, 3, 1, 1, 3, 7, 7, 1, 1, 3, 15, 1, 31, 63, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 3, 1, 7, 15, 1, 1, 1, 3, 1, 1, 3, 7, 3, 1, 1, 3, 7, 1, 15, 31, 3, 1, 1, 3, 1

Offset: 0

N. J. A. Sloane, Dec 22 2003

			13 = 1101 so a(13) = 1.

Cf. A089309-A089313.

a(2n+1) = A038712(n+1).

rb1[n_]:=Module[{id=Split[IntegerDigits[n,2]]},If[MemberQ[ Last[ id],0], FromDigits[ id[[-2]],2], FromDigits[id[[-1]],2]]]; Join[{0}, Array[ rb1,100]] (* Harvey P. Dale, Dec 18 2015 *)

More terms from Vladeta Jovovic, Jan 20 2004

A100892 a(n) = (2n-1) XOR (2n+1), bitwise.

Original entry on oeis.org

2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2, 6, 2, 62, 2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2, 6, 2, 126, 2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2, 6, 2, 62, 2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2, 6, 2, 254, 2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2, 6, 2, 62, 2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2

Offset: 1

Reinhard Zumkeller, Jan 10 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006519, A099820, A007088.

Cf. A005408.

a100892 n = (2 * n - 1) `xor` (2 * n + 1)
a100892_list = zipWith xor (tail a005408_list) a005408_list
-- Reinhard Zumkeller, Sep 03 2013

a[n_]:=BitXor[2*n-1,2*n+1]; a/@Range[100] (* Ivan N. Ianakiev, Jul 04 2019 *)

a(n)=4*2^valuation(n,2)-2; \\ Ralf Stephan, Aug 21 2013

def A100892(n): return ((~n& n-1)<<2)+2 # Chai Wah Wu, Jul 07 2022

a(n) = 2 * ((n-1) XOR n) = 2*A038712(n).
a(n) = 4*2^A007814(n) - 2.
Recurrence: a(2n) = 2a(n) + 2, a(2n+1) = 2. - Ralf Stephan, Aug 21 2013
a(n) = A088837(n) - 1. - Filip Zaludek, Dec 10 2016
a(n) = A074400(n)/A000593(n) = 2*A000203(n)/A000593(n). - Ivan N. Ianakiev, Jul 04 2019

A126851 SPM4Sigma(n) = (-1)^(1/2((Sum_i p_i)-Omega(m'))Sum_{d|n} (-1)^(1/2((Sum_j p_j)-Omega(d'))d =(2^(r+1)-1)Product_i [Sum_{1<=s_i<=r_i} p_i^s_i +(-1)^((p_i-1)/2)] where n=2^rm', gcd(2,m')=1, m'=Product_i p_i^r_i, d=2^k*d', gcd(2,d')=1, d'=Product_j p_j^r_j SPM4 for Signed by Prime factors Mod 4.

Original entry on oeis.org

1, 3, 2, 7, 6, 6, 6, 15, 11, 18, 10, 14, 14, 18, 12, 31, 18, 33, 18, 42, 12, 30, 22, 30, 31, 42, 38, 42, 30, 36, 30, 63, 20, 54, 36, 77, 38, 54, 28, 90, 42, 36, 42, 70, 66, 66, 46, 62, 55, 93, 36, 98, 54, 114, 60, 90, 36, 90, 58, 84, 62, 90, 66, 127, 84, 60, 66, 126, 44, 108, 70, 165, 74, 114, 62, 126, 60, 84, 78, 186

Offset: 1

Yasutoshi Kohmoto, Feb 24 2007

The name contains an unmatched parenthesis. - Editors, Mar 12 2024

			SPM4Sigma(240) = (1+2+4+8+16)*(-1+3)*(1+5).

Cf. A126852.

A126851 := proc(n)
    local r,mprime,piri,iprod,pi,ri,si;
    r := A007814(n) ;
    mprime := n/2^r ;
    iprod := 1 ;
    if mprime > 1 then
        for piri in ifactors(mprime)[2] do
            pi := op(1,piri) ;
            ri := op(2,piri) ;
            add(pi^si,si=1..ri) + (-1)^( (pi-1)/2) ;;
            iprod := iprod*% ;
        end do:
    end if;
    %*A038712(n) ;
end proc:
seq(A126851(n),n=1..40) ; # R. J. Mathar, Mar 13 2024

SPM4Sigma(n) = (2^r-1)*Product_i (p_i^(r_i+1)-p_i)/(p_i-1)+(-1)^(1/2*(p_i-1)) = (2^r-1)*Product_{i=1 mod 4} ((p_i^(r_i+1)-p_i)/(p_i-1)+1)*Product_{i=3 mod 4} ((p_i^(r_i+1)-p_i)/(p_i-1)-1)
a(2^n) = A000225(n+1). - R. J. Mathar, Mar 13 2024
A038712(n) | a(n). - R. J. Mathar, Mar 13 2024

a(2) and a(7) corrected, sequence extended beyond a(20). - R. J. Mathar, Mar 13 2024

A228349 Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Aug 26 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.

The equivalent sequence for partitions is A220482.

			----------------------------------------------------------
.             Diagram                Triangle
Compositions    of            of compositions (rows)
of 5          regions          and regions (columns)
----------------------------------------------------------
.            _ _ _ _ _
5           |_        |                                 5
1+4         |_|_      |                               1 4
2+3         |_  |     |                             2   3
1+1+3       |_|_|_    |                           1 1   3
3+2         |_    |   |                         3       2
1+2+2       |_|_  |   |                       1 2       2
2+1+2       |_  | |   |                     2   1       2
1+1+1+2     |_|_|_|_  |                   1 1   1       2
4+1         |_      | |                 4               1
1+3+1       |_|_    | |               1 3               1
2+2+1       |_  |   | |             2   2               1
1+1+2+1     |_|_|_  | |           1 1   2               1
3+1+1       |_    | | |         3       1               1
1+2+1+1     |_|_  | | |       1 2       1               1
2+1+1+1     |_  | | | |     2   1       1               1
1+1+1+1+1   |_|_|_|_|_|   1 1   1       1               1
.
Written as an irregular triangle in which row n lists the parts of the n-th region the sequence begins:
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,2,2,3,4;
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
...
Alternative interpretation of this sequence:
Triangle read by rows in which row r lists the regions of the last section of the set of compositions of r:
[1];
[1,2];
[1],[1,1,2,3];
[1],[1,2],[1],[1,1,1,1,2,2,3,4];
[1],[1,2],[1],[1,1,2,3],[1],[1,2],[1],[1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5];

Michael De Vlieger, Table of n, a(n) for n = 1..13312 (rows 1 <= n <= 2^11 = 2048).

Main triangle: Right border gives A001511. Row j has length A006519(j). Row sums give A038712.

Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A090996, A186114, A187816, A187818, A206437, A220482, A228347, A228348, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

Table[Map[Length@ TakeWhile[IntegerDigits[#, 2], # == 1 &] &, Range[2^(# - 1), 2^# - 1]] &@ IntegerExponent[2 n, 2], {n, 32}] // Flatten (* Michael De Vlieger, May 23 2017 *)

A342639 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = g(f(n) + f(k)) where g maps the complement, say s, of a finite set of nonnegative integers to the value Sum_{e >= 0 not in s} 2^e, f is the inverse of g, and "+" denotes the Minkowski sum.

Original entry on oeis.org

0, 1, 1, 0, 3, 0, 3, 1, 1, 3, 0, 7, 2, 7, 0, 1, 1, 3, 3, 1, 1, 0, 3, 0, 15, 0, 3, 0, 7, 1, 5, 3, 3, 5, 1, 7, 0, 15, 2, 7, 0, 7, 2, 15, 0, 1, 1, 7, 3, 1, 1, 3, 7, 1, 1, 0, 3, 0, 31, 4, 11, 4, 31, 0, 3, 0, 3, 1, 1, 3, 7, 5, 5, 7, 3, 1, 1, 3, 0, 7, 2, 7, 0, 15, 6, 15, 0, 7, 2, 7, 0

Offset: 0

Rémy Sigrist, Mar 17 2021

In other words:
- we consider the set S of sets s of nonnegative integers whose complement is finite,
- the function g encodes the "missing integers" in binary:
          g(A001477 \ {1, 4}) = 2^1 + 2^4 = 18
- the function f is the inverse of g:
          f(42) = f(2^1 + 2^3 + 2^5) = A001477 \ {1, 3, 5},
- the Minkowski sum of two sets, say U and V, is the set of sums u+v where u belongs to U and v belongs to V,
- the Minkowski sum is stable over S,
- and T provides an encoding for this operation.
This sequence has connections with A067138; here we consider complements of finite sets of nonnegative integers, there finite sets of nonnegative integers.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6    7   8   9  10  11  12  13  14   15
  ---+------------------------------------------------------------------
    0|   0   1   0   3   0   1   0    7   0   1   0   3   0   1   0   15
    1|   1   3   1   7   1   3   1   15   1   3   1   7   1   3   1   31
    2|   0   1   2   3   0   5   2    7   0   1   2  11   0   5   2   15
    3|   3   7   3  15   3   7   3   31   3   7   3  15   3   7   3   63
    4|   0   1   0   3   0   1   4    7   0   1   0   3   0   9   4   15
    5|   1   3   5   7   1  11   5   15   1   3   5  23   1  11   5   31
    6|   0   1   2   3   4   5   6    7   0   9   2  11   4  13   6   15
    7|   7  15   7  31   7  15   7   63   7  15   7  31   7  15   7  127
    8|   0   1   0   3   0   1   0    7   0   1   0   3   0   1   8   15
    9|   1   3   1   7   1   3   9   15   1   3   1   7   1  19   9   31
   10|   0   1   2   3   0   5   2    7   0   1  10  11   0   5  10   15
   11|   3   7  11  15   3  23  11   31   3   7  11  47   3  23  11   63
   12|   0   1   0   3   0   1   4    7   0   1   0   3   8   9  12   15
   13|   1   3   5   7   9  11  13   15   1  19   5  23   9  27  13   31
   14|   0   1   2   3   4   5   6    7   8   9  10  11  12  13  14   15
   15|  15  31  15  63  15  31  15  127  15  31  15  63  15  31  15  255

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Wikipedia, Minkowski addition

Cf. A038712, A067138, A133457, A135481, A342640, A342641, A342642.

T(n,k) = { my (v=0); for (x=0, #binary(n)+#binary(k), my (f=0); for (y=0, x, if (!bittest(n,y) && !bittest(k,x-y), f=1; break)); if (!f, v+=2^x)); return (v) }

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 0) = A135481(n).
T(n, 1) = A038712(n+1).
T(2^n-1, 2^k-1) = 2^(n+k)-1.
T(n, n) = A342640(n).

A378988 a(n) = 2*n XOR 1+sigma(n), where XOR is bitwise-xor, A003987.

Original entry on oeis.org

0, 0, 3, 0, 13, 1, 7, 0, 28, 7, 27, 5, 21, 5, 7, 0, 49, 12, 51, 3, 11, 9, 55, 13, 18, 31, 31, 1, 37, 117, 31, 0, 115, 115, 119, 20, 109, 113, 119, 11, 121, 53, 123, 13, 21, 21, 111, 29, 88, 58, 47, 11, 93, 21, 39, 9, 35, 47, 75, 209, 69, 29, 23, 0, 215, 21, 195, 247, 235, 29, 199, 84, 217, 231, 235, 21, 251, 53, 207

Offset: 1

Antti Karttunen, Dec 16 2024

For any hypothetical quasiperfect number q (for which sigma(q) = 2*q+1, see A336701), a(q) would be equal to 2*q XOR 2*q+2 = 2*(q XOR q+1) = 2*A038712(1+q) = A100892(1+q).

See also A000079 and A235796 concerning the "almost perfect" or "least deficient" numbers that give positions of 0's here.

Cf. A000079 (conjectured to give positions of all 0's), A000396 (positions of 1's), A000203, A003987, A028982 (positions of even terms), A028983 (of odd terms), A038712, A100892, A318467, A336701, A378998, A379009 [= a(n^2)].

Cf. also A235796, A294898, A294899, A318457.

Array[BitXor[2*#, DivisorSigma[1, #] + 1] &, 100] (* Paolo Xausa, Dec 16 2024 *)

PARI
```
A378988(n) = bitxor(n+n,1+sigma(n));
```

For all n in A028983, a(n) = 2n+1 XOR sigma(n) = 1+A318467(n).

cp's OEIS Frontend

A324056 a(n) = A000593(A005940(1+n)).

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A326988 Sum of nonpowers of 2 dividing n.

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A355584 a(n) is the sum of the 5-smooth divisors of n.

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A088841 Numerator of the quotient sigma(7*n)/sigma(n).

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A089312 Write n in binary; a(n) = number represented by rightmost block of 1's.

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A100892 a(n) = (2*n-1) XOR (2*n+1), bitwise.

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A228349 Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

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A100892 a(n) = (2n-1) XOR (2n+1), bitwise.