A014311 -id:A014311 - cp's OEIS frontend

A000741 Number of compositions of n into 3 ordered relatively prime parts.

0, 0, 1, 3, 6, 9, 15, 18, 27, 30, 45, 42, 66, 63, 84, 84, 120, 99, 153, 132, 174, 165, 231, 180, 270, 234, 297, 270, 378, 276, 435, 360, 450, 408, 540, 414, 630, 513, 636, 552, 780, 558, 861, 690, 828, 759, 1035, 744, 1113, 870, 1104, 972, 1326, 945, 1380, 1116, 1386, 1218

Offset: 1

N. J. A. Sloane

			From _Gus Wiseman_, Oct 14 2020: (Start)
The a(3) = 1 through a(8) = 18 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,2,4)  (1,2,5)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,3,3)  (1,3,4)
                    (2,1,2)  (1,4,1)  (1,4,2)  (1,4,3)
                    (2,2,1)  (2,1,3)  (1,5,1)  (1,5,2)
                    (3,1,1)  (2,3,1)  (2,1,4)  (1,6,1)
                             (3,1,2)  (2,2,3)  (2,1,5)
                             (3,2,1)  (2,3,2)  (2,3,3)
                             (4,1,1)  (2,4,1)  (2,5,1)
                                      (3,1,3)  (3,1,4)
                                      (3,2,2)  (3,2,3)
                                      (3,3,1)  (3,3,2)
                                      (4,1,2)  (3,4,1)
                                      (4,2,1)  (4,1,3)
                                      (5,1,1)  (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
N. J. A. Sloane, Transforms

A000010 is the length-2 version.
A000217(n-2) does not require relative primality.
A000740 counts these compositions of any length.
A000742 is the length-4 version.
A000837 counts relatively prime partitions.
A023023 is the unordered version.
A101271 is the strict case.
A101391 has this as column k = 3.
A284825*6 is the pairwise non-coprime case.
A291166 intersected with A014311 ranks these compositions.
A337461 is the pairwise coprime instead of relatively prime version.
A337603 counts length-3 compositions whose distinct parts are pairwise coprime.
A337604 is the pairwise non-coprime instead of relatively prime version.
Cf. A001399, A007997, A023022, A078374, A337450, A337451, A337602.

with(numtheory):
mobtr:= proc(p)
          proc(n) option remember;
            add(mobius(n/d)*p(d), d=divisors(n))
          end
        end:
A000217:= n-> n*(n+1)/2:
a:= mobtr(n-> A000217(n-2)):
seq(a(n), n=1..58);  # Alois P. Heinz, Feb 08 2011

mobtr[p_] := Module[{f}, f[n_] := f[n] = Sum[MoebiusMu[n/d]*p[d], {d, Divisors[n]}]; f]; A000217[n_] := n*(n+1)/2; a = mobtr[A000217[#-2]&]; Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Mar 12 2014, after Alois P. Heinz *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],GCD@@#==1&]],{n,0,30}] (* Gus Wiseman, Oct 14 2020 *)

Moebius transform of A000217(n-2).

G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = (1 - 3*x + 3*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 26 2017

Edited by Alois P. Heinz, Feb 08 2011

A023688 Numbers with exactly 6 ones in binary expansion.

Original entry on oeis.org

63, 95, 111, 119, 123, 125, 126, 159, 175, 183, 187, 189, 190, 207, 215, 219, 221, 222, 231, 235, 237, 238, 243, 245, 246, 249, 250, 252, 287, 303, 311, 315, 317, 318, 335, 343, 347, 349, 350, 359, 363, 365, 366, 371, 373

Offset: 1

Olivier Gérard

Sequence appears to include all numbers m such that 8^5 is the highest power of 2 dividing A005148(m). General conjecture: numbers k such that 8^j is the highest power of 2 dividing A005148(k) is the same sequence as numbers having exactly (j+1) 1's in their binary representation. - Benoit Cloitre, Jun 22 2002

Ivan Neretin, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Cf. A000079, A018900, A014311, A014312, A014313, A023689, A023690, A023691 (Hamming weight = 1..9).

Cf. A005148, A057168.

Select[ Range[ 63, 380 ], (Count[ IntegerDigits[ #, 2 ], 1 ]==6)& ]

is_A023688(n)=hammingweight(n)==6 \\ M. F. Hasler, Aug 27 2014

print1(t=2^6-1); for(i=2, 50, print1(", "t=A057168(t))) \\ M. F. Hasler, Aug 27 2014

from itertools import islice
def A023688_gen(): # generator of terms
    yield (n:=63)
    while True: yield (n:=((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A023688_list = list(islice(A023688_gen(),30)) # Chai Wah Wu, Mar 06 2025

a(n+1) = A057168(a(n)). - M. F. Hasler, Aug 27 2014

Sum_{n>=1} 1/a(n) = 1.387753111935705074750004158584017188750706394077047633137401652680870607884... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

A023691 Numbers with exactly 9 ones in binary expansion.

Original entry on oeis.org

511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1279, 1407, 1471, 1503, 1519, 1527, 1531, 1533, 1534, 1663, 1727, 1759, 1775, 1783, 1787, 1789, 1790, 1855, 1887, 1903, 1911, 1915, 1917, 1918, 1951, 1967, 1975

Offset: 1

Olivier Gérard

Ivan Neretin, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690 (Hamming weight = 1, 2, ..., 8), A057168.

Select[ Range[ 511, 2048 ], (Count[ IntegerDigits[ #, 2 ], 1 ]==9)& ]

is_A023691(n)=hammingweight(n)==9 \\ M. F. Hasler, Aug 27 2014

print1(t=2^9-1); for(i=2, 50, print1(", "t=A057168(t))) \\ M. F. Hasler, Aug 27 2014

from itertools import islice
def A023691_gen(): # generator of terms
    yield (n:=511)
    while True: yield (n:=((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A023691_list = list(islice(A023691_gen(),30)) # Chai Wah Wu, Mar 06 2025

a(n+1) = A057168(a(n)). - M. F. Hasler, Aug 27 2014

Sum_{n>=1} 1/a(n) = 1.386348105265697723853732153017686670695581836109569165990080192653647019078... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

A081091 Primes of the form 2^i + 2^j + 1, i > j > 0.

Original entry on oeis.org

7, 11, 13, 19, 37, 41, 67, 73, 97, 131, 137, 193, 521, 577, 641, 769, 1033, 1153, 2053, 2081, 2113, 4099, 4129, 8209, 12289, 16417, 18433, 32771, 32801, 32833, 40961, 65539, 133121, 147457, 163841, 262147, 262153, 262657, 270337, 524353, 524801

Offset: 1

Reinhard Zumkeller, Mar 05 2003

This is sequence A070739 without the Fermat primes, A000215. Sequence A081504 lists the i for which there are no primes. - T. D. Noe, Jun 22 2007

Primes in A014311. - Reinhard Zumkeller, May 03 2012

			    7 = 2^2 + 2^1 + 1
   11 = 2^3 + 2^1 + 1
   13 = 2^3 + 2^2 + 1
   19 = 2^4 + 2^1 + 1
   37 = 2^5 + 2^2 + 1
   41 = 2^5 + 2^3 + 1
   67 = 2^6 + 2^1 + 1
   73 = 2^6 + 2^3 + 1
   97 = 2^6 + 2^5 + 1
  131 = 2^7 + 2^1 + 1
  137 = 2^7 + 2^3 + 1
  193 = 2^7 + 2^6 + 1
  521 = 2^9 + 2^3 + 1

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..7800 (n = 1..1000 from T. D. Noe)
Richard Ehrenborg and N. Bradley Fox, The Descent Set Polynomial Revisited, arXiv:1408.6858 [math.CO], 2014.
Norman B. Fox, Combinatorial Potpourri: Permutations, Products, Posets, and Pfaffians, University of Kentucky, Theses and Dissertations, Mathematics, Paper 25.

Essentially the same as A070739.
Cf. A000040, A000215, A081092, A010051.
Cf. A095077 (primes with four bits set).
A057733 = 2^A057732 + 3 and A039687 = 3*2^A002253 + 1 are subsequences.

a081091 n = a081091_list !! (n-1)
a081091_list = filter ((== 1) . a010051') a014311_list
-- Reinhard Zumkeller, May 03 2012

N:= 20: # to get all terms < 2^N
select(isprime, [seq(seq(2^i+2^j+1,j=1..i-1),i=1..N-1)]); # Robert Israel, May 17 2016

Select[Flatten[Table[2^i + 2^j + 1, {i, 21}, {j, i-1}]], PrimeQ] (* Alonso del Arte, Jan 11 2011 *)

do(mx)=my(v=List(),t); for(i=2,mx,for(j=1,i-1,if(ispseudoprime(t=2^i+2^j+1), listput(v,t)))); Vec(v) \\ Charles R Greathouse IV, Jan 02 2014

is(n)=hammingweight(n)==3 && isprime(n) \\ Charles R Greathouse IV, Aug 28 2017

A81091=[7]; next_A081091(p, i=exponent(p), j=exponent(p-2^i))=!until(isprime(2^i+2^j+1), j++>=i && i++ && j=1)+2^i+2^j
A081091(n)={for(k=#A81091, n-1, A81091=concat(A81091, next_A081091(A81091[k]))); A81091[n]} \\ M. F. Hasler, Mar 03 2023

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A081091_gen(): # generator of terms
    return filter(isprime,map(lambda s:int('1'+''.join(s)+'1',2),(s for l in count(1) for s in multiset_permutations('0'*(l-1)+'1'))))
A081091_list = list(islice(A081091_gen(),30)) # Chai Wah Wu, Jul 19 2022

A000120(a(n)) = 3.

A156040 Number of compositions (ordered partitions) of n into 3 parts (some of which may be zero), where the first is at least as great as each of the others.

Original entry on oeis.org

1, 1, 3, 4, 6, 8, 11, 13, 17, 20, 24, 28, 33, 37, 43, 48, 54, 60, 67, 73, 81, 88, 96, 104, 113, 121, 131, 140, 150, 160, 171, 181, 193, 204, 216, 228, 241, 253, 267, 280, 294, 308, 323, 337, 353, 368, 384, 400, 417, 433, 451, 468, 486, 504, 523, 541, 561, 580, 600

Offset: 0

Jack W Grahl, Feb 02 2009, Feb 11 2009

For n = 1, 2 these are just the triangular numbers. a(n) is always at least 1/3 of the corresponding triangular number, since each partition of this type gives up to three ordered partitions with the same cyclical order.
An alternative definition, which avoids using parts of size 0: a(n) is the third diagonal of A184957. - N. J. A. Sloane, Feb 27 2011
Diagonal sums of the triangle formed by rows T(2, k) k = 0, 1, ..., 2m of ascending m-nomial triangles (see A004737):
1
1 2 1
1 2 3 2 1
1 2 3 4 3 2 1
1 2 3 4 5 4 3 2 1
1 2 3 4 5 6 5 4 3 2 1
- Bob Selcoe, Feb 07 2014
Arrange A004396 in rows successively shifted to the right two spaces and sum the columns:
1  1  2  3  3  4  5  5  6 ...
      1  1  2  3  3  4  5 ...
            1  1  2  3  3 ...
                  1  1  2 ...
                        1 ...
------------------------------
1  1  3  4  6  8 11 13 17 ... - L. Edson Jeffery, Jul 30 2014
a(n) is the dimension of three-dimensional (2n + 2)-homogeneous polynomial vector fields with full tetrahedral symmetry (for a given orthogonal representation), and which are solenoidal. - Giedrius Alkauskas, Sep 30 2017
Also the number of compositions of n + 3 into three parts, the first at least as great as each of the other two. Also the number of compositions of n + 4 into three parts, the first strictly greater than each of the other two. - Gus Wiseman, Oct 09 2020

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + 20*x^9 + ...
The a(4) = 6 compositions of 4 are: (4 0 0), (3 1 0), (3 0 1), (2 2 0), (2 1 1), (2 0 2).
From _Gus Wiseman_, Oct 05 2020: (Start)
The a(0) = 1 through a(7) = 13 triples of nonnegative integers summing to n where the first is at least as great as each of the other two are:
  (000)  (100)  (101)  (111)  (202)  (212)  (222)  (313)
                (110)  (201)  (211)  (221)  (303)  (322)
                (200)  (210)  (220)  (302)  (312)  (331)
                       (300)  (301)  (311)  (321)  (403)
                              (310)  (320)  (330)  (412)
                              (400)  (401)  (402)  (421)
                                     (410)  (411)  (430)
                                     (500)  (420)  (502)
                                            (501)  (511)
                                            (510)  (520)
                                            (600)  (601)
                                                   (610)
                                                   (700)
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Giedrius Alkauskas, Projective and polynomial superflows. I, arxiv.org/1601.06570 [math.AG] (2017), Section 4.3.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

For compositions into 4 summands see A156039; also see A156041 and A156042.
Cf. A184957, A071619 (bisection).
A001399(n-2)*2 is the strict case.
A001840(n-2) is the version with opposite relations.
A001840(n-1) is the version with strict opposite relations.
A069905 is the case with strict relations.
A014311 ranks 3-part compositions, with strict case A337453.
A014612 ranks 3-part partitions, with strict case A007304.
Cf. A000217, A001523, A211540, A218004, A220377, A337483, A337484.

a:= proc(n) local m, r; m := iquo(n, 6, 'r'); (4 +6*m +2*r) *m + [1, 1, 3, 4, 6, 8][r+1] end: seq(a(n), n=0..60); # Alois P. Heinz, Jun 14 2009

nn = 58; CoefficientList[Series[x^3/(1 - x^2)^2/(1 - x^3) + 1/(1 - x^2)^2/(1 - x), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 14 2013 *)
CoefficientList[Series[(1 + x^2)/((1 + x) * (1 + x + x^2) * (1 - x)^3), {x, 0, 58}], x] (* L. Edson Jeffery, Jul 29 2014 *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 1, 3, 4, 6, 8}, 60] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+3,{3}],#[[1]]>=#[[2]]&&#[[1]]>=#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020*)

{a(n) = n*(n+4)\6 + 1}; /* Michael Somos, Mar 26 2017 */

G.f.: (x^2+1) / (1-x-x^2+x^4+x^5-x^6). - Alois P. Heinz, Jun 14 2009
Slightly nicer g.f.: (1+x^2)/((1-x)*(1-x^2)*(1-x^3)). - N. J. A. Sloane, Apr 29 2011
a(n) = A007590(n+2) - A000212(n+2). - Richard R. Forberg, Dec 08 2013
a(2*n) = A071619(n+1). - L. Edson Jeffery, Jul 29 2014
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 6, a(5) = 8. - Harvey P. Dale, May 28 2015
a(n) = (n^2 + 4*n + 3)/6 + IF(MOD(n, 2) = 0, 1/2) + IF(MOD(n, 3) = 1, -1/3). - Heinrich Ludwig, Mar 21 2017
a(n) = 1 + floor((n^2 + 4*n)/6). - Giovanni Resta, Mar 21 2017
Euler transform of length 4 sequence [1, 2, 1, -1]. - Michael Somos, Mar 26 2017
a(n) = a(-4 - n) for all n in Z. - Michael Somos, Mar 26 2017
0 = a(n)*(-1 + a(n) - 2*a(n+1) - 2*a(n+2) + 2*a(n+3)) + a(n+1)*(+1 + a(n+1) + 2*a(n+2) - 2*a(n+3)) + a(n+2)*(+1 + a(n+2) - 2*a(n+3)) + a(n+3)*(-1 + a(n+3)) for all n in Z. - Michael Somos, Mar 26 2017
a(n) = round((n+1)*(n+3)/6). - Bill McEachen, Feb 16 2021
Sum_{n>=0} 1/a(n) = 3/2 + Pi^2/36 + (tan(c1)-1)*c1 + 3*c2*sinh(c2)/(1+2*cosh(c2)), where c1 = Pi/(2*sqrt(3)) and c2 = Pi*sqrt(2)/3. - Amiram Eldar, Dec 10 2022
E.g.f.: ((16 + 15*x + 3*x^2)*cosh(x) + 2*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + (7 + 15*x + 3*x^2)*sinh(x))/18. - Stefano Spezia, Apr 05 2023

More terms from Alois P. Heinz, Jun 14 2009

A337603 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 9, 9, 18, 15, 24, 21, 42, 24, 51, 30, 54, 42, 93, 45, 102, 54, 99, 69, 162, 66, 150, 87, 168, 96, 264, 93, 228, 120, 246, 126, 336, 132, 315, 168, 342, 162, 486, 165, 420, 216, 411, 213, 618, 207, 558, 258, 540, 258, 783, 264, 654, 324, 660

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

			The a(3) = 1 through a(8) = 18 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,3,3)  (1,2,5)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,5,1)  (1,3,4)
                    (2,1,2)  (1,4,1)  (2,2,3)  (1,4,3)
                    (2,2,1)  (2,1,3)  (2,3,2)  (1,5,2)
                    (3,1,1)  (2,3,1)  (3,1,3)  (1,6,1)
                             (3,1,2)  (3,2,2)  (2,1,5)
                             (3,2,1)  (3,3,1)  (2,3,3)
                             (4,1,1)  (5,1,1)  (2,5,1)
                                               (3,1,4)
                                               (3,2,3)
                                               (3,3,2)
                                               (3,4,1)
                                               (4,1,3)
                                               (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A014311 intersected with A333228 ranks these compositions.
A220377*6 is the strict case.
A337461 is the strict case except for any number of 1's.
A337601 is the unordered version.
A337602 considers all singletons to be coprime.
A337665 counts these compositions of any length, ranked by A333228 with complement A335238.
A000217(n - 2) counts 3-part compositions.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A007318 and A097805 count compositions by length.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A304711 ranks partitions whose distinct parts are pairwise coprime.
A305713 counts strict pairwise coprime partitions.
A327516 counts pairwise coprime partitions, with strict case A305713.
A333227 ranks pairwise coprime compositions.
Cf. A000740, A001840, A007359, A087087, A178472, A284825, A302696, A307719, A335235, A337561, A337695.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],CoprimeQ@@Union[#]&]],{n,0,100}]

A337453 Numbers k such that the k-th composition in standard order is an ordered triple of distinct positive integers.

Original entry on oeis.org

37, 38, 41, 44, 50, 52, 69, 70, 81, 88, 98, 104, 133, 134, 137, 140, 145, 152, 161, 176, 194, 196, 200, 208, 261, 262, 265, 268, 274, 276, 289, 290, 296, 304, 321, 324, 328, 352, 386, 388, 400, 416, 517, 518, 521, 524, 529, 530, 532, 536, 545, 560, 577, 578

Offset: 1

Gus Wiseman, Sep 07 2020

nonn

The k-th composition in standard order (graded reverse-lexicographic, 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. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding triples begins:
     37: (3,2,1)    140: (4,1,3)    289: (3,5,1)
     38: (3,1,2)    145: (3,4,1)    290: (3,4,2)
     41: (2,3,1)    152: (3,1,4)    296: (3,2,4)
     44: (2,1,3)    161: (2,5,1)    304: (3,1,5)
     50: (1,3,2)    176: (2,1,5)    321: (2,6,1)
     52: (1,2,3)    194: (1,5,2)    324: (2,4,3)
     69: (4,2,1)    196: (1,4,3)    328: (2,3,4)
     70: (4,1,2)    200: (1,3,4)    352: (2,1,6)
     81: (2,4,1)    208: (1,2,5)    386: (1,6,2)
     88: (2,1,4)    261: (6,2,1)    388: (1,5,3)
     98: (1,4,2)    262: (6,1,2)    400: (1,3,5)
    104: (1,2,4)    265: (5,3,1)    416: (1,2,6)
    133: (5,2,1)    268: (5,1,3)    517: (7,2,1)
    134: (5,1,2)    274: (4,3,2)    518: (7,1,2)
    137: (4,3,1)    276: (4,2,3)    521: (6,3,1)

6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts these compositions.
A007304 is an unordered version.
A014311 is the non-strict version.
A337461 counts the coprime case.
A000217(n - 2) counts 3-part compositions.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.
A014612 ranks 3-part partitions.
Cf. A000212, A220377, A307534, A337459, A337460, A337561, A337603, A337604.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Length[stc[#]]==3&&UnsameQ@@stc[#]&]

These triples are counted by 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1).

Intersection of A014311 and A233564.

A337484 Number of ordered triples of positive integers summing to n that are neither strictly increasing nor strictly decreasing.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 8, 13, 17, 22, 28, 35, 41, 50, 58, 67, 77, 88, 98, 111, 123, 136, 150, 165, 179, 196, 212, 229, 247, 266, 284, 305, 325, 346, 368, 391, 413, 438, 462, 487, 513, 540, 566, 595, 623, 652, 682, 713, 743, 776, 808, 841, 875, 910, 944, 981, 1017

Offset: 0

Gus Wiseman, Sep 11 2020

nonn

			The a(3) = 1 through a(7) = 13 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)
           (1,2,1)  (1,2,2)  (1,3,2)  (1,3,3)
           (2,1,1)  (1,3,1)  (1,4,1)  (1,4,2)
                    (2,1,2)  (2,1,3)  (1,5,1)
                    (2,2,1)  (2,2,2)  (2,1,4)
                    (3,1,1)  (2,3,1)  (2,2,3)
                             (3,1,2)  (2,3,2)
                             (4,1,1)  (2,4,1)
                                      (3,1,3)
                                      (3,2,2)
                                      (3,3,1)
                                      (4,1,2)
                                      (5,1,1)

A140106 is the unordered case.
A242771 allows strictly increasing but not strictly decreasing triples.
A337481 counts these compositions of any length.
A001399(n - 6) counts unordered strict triples.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A069905 counts unordered triples.
A218004 counts strictly increasing or weakly decreasing compositions.
A332745 counts partitions with weakly increasing or weakly decreasing run-lengths.
A332835 counts compositions with weakly increasing or weakly decreasing run-lengths.
A337483 counts triples either weakly increasing or weakly decreasing.
Cf. A000212, A000217, A001840, A014311, A046691, A128422, A156040, A332834, A337461, A337482, A337561, A337603, A337604.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&&!Greater@@#&]],{n,0,15}]

a(n) = 2*A242771(n - 1) - A000217(n - 1), n > 0.
2*A001399(n - 6) = 2*A069905(n - 3) = 2*A211540(n - 1) is the complement.
4*A001399(n - 6) = 4*A069905(n - 3) = 4*A211540(n - 1) is the strict case.
Conjectures from Colin Barker, Sep 13 2020: (Start)
G.f.: x^3*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>6.
(End)

A128422 Projective plane crossing number of K_{4,n}.

Original entry on oeis.org

0, 0, 0, 2, 4, 6, 10, 14, 18, 24, 30, 36, 44, 52, 60, 70, 80, 90, 102, 114, 126, 140, 154, 168, 184, 200, 216, 234, 252, 270, 290, 310, 330, 352, 374, 396, 420, 444, 468, 494, 520, 546, 574, 602, 630, 660, 690, 720, 752, 784, 816, 850, 884, 918, 954, 990, 1026

Offset: 1

Eric W. Weisstein, Mar 02 2007

From Gus Wiseman, Oct 15 2020: (Start)
Also the number of 3-part compositions of n that are neither strictly increasing nor weakly decreasing. The set of numbers k such that row k of A066099 is such a composition is the complement of A333255 (strictly increasing) and A114994 (weakly decreasing) in A014311 (triples). The a(4) = 2 through a(9) = 14 compositions are:
  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)  (1,1,7)
  (1,2,1)  (1,2,2)  (1,3,2)  (1,3,3)  (1,4,3)  (1,4,4)
           (1,3,1)  (1,4,1)  (1,4,2)  (1,5,2)  (1,5,3)
           (2,1,2)  (2,1,3)  (1,5,1)  (1,6,1)  (1,6,2)
                    (2,3,1)  (2,1,4)  (2,1,5)  (1,7,1)
                    (3,1,2)  (2,2,3)  (2,2,4)  (2,1,6)
                             (2,3,2)  (2,3,3)  (2,2,5)
                             (2,4,1)  (2,4,2)  (2,4,3)
                             (3,1,3)  (2,5,1)  (2,5,2)
                             (4,1,2)  (3,1,4)  (2,6,1)
                                      (3,2,3)  (3,1,5)
                                      (3,4,1)  (3,2,4)
                                      (4,1,3)  (3,4,2)
                                      (5,1,2)  (3,5,1)
                                               (4,1,4)
                                               (4,2,3)
                                               (5,1,3)
                                               (6,1,2)
(End)

Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Projective Plane Crossing Number
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

A007997 counts the complement.
A337482 counts these compositions of any length.
A337484 is the non-strict/non-strict version.
A000009 counts strictly increasing compositions, ranked by A333255.
A000041 counts weakly decreasing compositions, ranked by A114994.
A001523 counts unimodal compositions (strict: A072706).
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A225620 ranks weakly increasing compositions.
A333149 counts neither increasing nor decreasing strict compositions.
A333256 ranks strictly decreasing compositions.
A337483 counts 3-part weakly increasing or weakly decreasing compositions.
Cf. A000217, A001399, A001840, A059204, A072705, A115981, A329398, A332578, A332831, A332833, A332834, A332874, A333147, A333190.

Table[Floor[((n - 2)^2 + (n - 2))/3], {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)
Table[Ceiling[n^2/3] - n, {n, 20}] (* Eric W. Weisstein, Sep 07 2018 *)
Table[(3 n^2 - 9 n + 4 - 4 Cos[2 n Pi/3])/9, {n, 20}] (* Eric W. Weisstein, Sep 07 2018 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 0, 0, 2, 4, 6}, 20] (* Eric W. Weisstein, Sep 07 2018 *)
CoefficientList[Series[-2 x^3/((-1 + x)^3 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 07 2018 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&&!GreaterEqual@@#&]],{n,15}] (* Gus Wiseman, Oct 15 2020 *)

a(n)=(n-1)*(n-2)\3 \\ Charles R Greathouse IV, Jun 06 2013

a(n) = floor(n/3)*(2n-3(floor(n/3)+1)).
a(n) = ceiling(n^2/3) - n. - Charles R Greathouse IV, Jun 06 2013
G.f.: -2*x^4 / ((x-1)^3*(x^2+x+1)). - Colin Barker, Jun 06 2013
a(n) = floor((n - 1)(n - 2) / 3). - Christopher Hunt Gribble, Oct 13 2009
a(n) = 2*A001840(n-3). - R. J. Mathar, Jul 21 2015
a(n) = A000217(n-2) - A001399(n-6) - A001399(n-3). - Gus Wiseman, Oct 15 2020
Sum_{n>=4} 1/a(n) = 10/3 - Pi/sqrt(3). - Amiram Eldar, Sep 27 2022

A263017 n is the a(n)-th positive integer having its binary weight.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 1, 4, 4, 5, 2, 6, 3, 4, 1, 5, 7, 8, 5, 9, 6, 7, 2, 10, 8, 9, 3, 10, 4, 5, 1, 6, 11, 12, 11, 13, 12, 13, 6, 14, 14, 15, 7, 16, 8, 9, 2, 15, 17, 18, 10, 19, 11, 12, 3, 20, 13, 14, 4, 15, 5, 6, 1, 7, 16, 17, 21, 18, 22, 23, 16, 19, 24, 25, 17

Offset: 1

Paul Tek, Oct 07 2015

Binary weight is given by A000120.
a(2^k) = k+1 for any k>=0.
a(2^k-1) = 1 for any k>0.
a(A057168(k)) = a(k)+1 for any k>0.
a(A036563(k+1)) = k for any k>0.
Ordinal transform of A000120. - Alois P. Heinz, Dec 23 2018

			The numbers with binary weight 3 are: 7, 11, 13, 14, 19, ...
Hence: a(7)=1, a(11)=2, a(13)=3, a(14)=4, a(19)=5, ...
And more generally: a(A014311(k))=k for any k>0.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, PERL program for this sequence

One more than A068076.
Cf. A000120, A036563, A057168, A254524, A286478, A286552, A286554, A286558.
Cf. A263109, A263110.

import Data.IntMap (empty, findWithDefault, insert)
a263017 n = a263017_list !! (n-1)
a263017_list = f 1 empty where
   f x m = y : f (x + 1) (insert h (y + 1) m) where
           y = findWithDefault 1 h m
           h = a000120 x
-- Reinhard Zumkeller, Oct 09 2015

a:= proc() option remember; local a, b, t; b, a:=
      proc() 0 end, proc(n) option remember; a(n-1);
        t:= add(i, i=convert(n, base, 2)); b(t):= b(t)+1
      end; a(0):=0; a
    end():
seq(a(n), n=1..120);  # Alois P. Heinz, Dec 23 2018

Perl
```
# See Links section.
```

from math import comb
def A263017(n):
    c, k = 1, 0
    for i,j in enumerate(bin(n)[-1:1:-1]):
        if j == '1':
            k += 1
            c += comb(i,k)
    return c # Chai Wah Wu, Mar 01 2023

a(n) = 1 + A068076(n). - Antti Karttunen, May 22 2017

cp's OEIS Frontend

A000741 Number of compositions of n into 3 ordered relatively prime parts.

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A023688 Numbers with exactly 6 ones in binary expansion.

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A023691 Numbers with exactly 9 ones in binary expansion.

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A081091 Primes of the form 2^i + 2^j + 1, i > j > 0.

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A156040 Number of compositions (ordered partitions) of n into 3 parts (some of which may be zero), where the first is at least as great as each of the others.

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A337603 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is not considered coprime unless it is (1).

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A337453 Numbers k such that the k-th composition in standard order is an ordered triple of distinct positive integers.