A291166 -id:A291166 - cp's OEIS frontend

A359495 Sum of positions of 1's in binary expansion minus sum of positions of 1's in reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

0, 0, -1, 0, -2, 0, -2, 0, -3, 0, -2, 1, -4, -1, -3, 0, -4, 0, -2, 2, -4, 0, -2, 2, -6, -2, -4, 0, -6, -2, -4, 0, -5, 0, -2, 3, -4, 1, -1, 4, -6, -1, -3, 2, -5, 0, -2, 3, -8, -3, -5, 0, -7, -2, -4, 1, -9, -4, -6, -1, -8, -3, -5, 0, -6, 0, -2, 4, -4, 2, 0, 6

Offset: 0

Gus Wiseman, Jan 05 2023

Also the sum of partial sums of reversed binary expansion minus sum of partial sums of binary expansion.

			The binary expansion of 158 is (1,0,0,1,1,1,1,0), with positions of 1's {1,4,5,6,7} with sum 23, reversed {2,3,4,5,8} with sum 22, so a(158) = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..16383

Indices of positive terms are A359401.
Indices of 0's are A359402.
A030190 gives binary expansion, reverse A030308.
A070939 counts binary digits.
A230877 adds up positions of 1's in binary expansion, reverse A029931.
Cf. A000120, A048793, A053632, A222955, A231204, A291166, A326669, A326672, A326673, A359042, A359043.

a:= n-> (l-> add(i*(l[-i]-l[i]), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..127);  # Alois P. Heinz, Jan 09 2023

sap[q_]:=Sum[q[[i]]*(2i-Length[q]-1),{i,Length[q]}];
Table[sap[IntegerDigits[n,2]],{n,0,100}]

def A359495(n):
    k = n.bit_length()-1
    return sum((i<<1)-k for i, j in enumerate(bin(n)[2:]) if j=='1') # Chai Wah Wu, Jan 09 2023

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

If n = Sum_{i=1..k} q_i * 2^(i-1), then a(n) = Sum_{i=1..k} q_i * (2i-k-1).

A335236 Numbers k such that the k-th composition in standard order (A066099) is not a singleton nor pairwise coprime.

Original entry on oeis.org

0, 10, 21, 22, 26, 34, 36, 40, 42, 43, 45, 46, 53, 54, 58, 69, 70, 73, 74, 76, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 98, 100, 104, 106, 107, 109, 110, 117, 118, 122, 130, 136, 138, 139, 141, 142, 146, 147, 148, 149, 150, 153, 154, 156, 160, 162, 163, 164

Offset: 1

Gus Wiseman, May 28 2020

nonn

These are compositions whose product is strictly greater than the LCM of their parts.

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 compositions begins:
    0: ()            74: (3,2,2)        109: (1,2,1,2,1)
   10: (2,2)         76: (3,1,3)        110: (1,2,1,1,2)
   21: (2,2,1)       81: (2,4,1)        117: (1,1,2,2,1)
   22: (2,1,2)       82: (2,3,2)        118: (1,1,2,1,2)
   26: (1,2,2)       84: (2,2,3)        122: (1,1,1,2,2)
   34: (4,2)         85: (2,2,2,1)      130: (6,2)
   36: (3,3)         86: (2,2,1,2)      136: (4,4)
   40: (2,4)         87: (2,2,1,1,1)    138: (4,2,2)
   42: (2,2,2)       88: (2,1,4)        139: (4,2,1,1)
   43: (2,2,1,1)     90: (2,1,2,2)      141: (4,1,2,1)
   45: (2,1,2,1)     91: (2,1,2,1,1)    142: (4,1,1,2)
   46: (2,1,1,2)     93: (2,1,1,2,1)    146: (3,3,2)
   53: (1,2,2,1)     94: (2,1,1,1,2)    147: (3,3,1,1)
   54: (1,2,1,2)     98: (1,4,2)        148: (3,2,3)
   58: (1,1,2,2)    100: (1,3,3)        149: (3,2,2,1)
   69: (4,2,1)      104: (1,2,4)        150: (3,2,1,2)
   70: (4,1,2)      106: (1,2,2,2)      153: (3,1,3,1)
   73: (3,3,1)      107: (1,2,2,1,1)    154: (3,1,2,2)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

The version for prime indices is A316438.
The version for binary indices is A335237.
The complement is A335235.
The version with singletons allowed is A335239.
Binary indices are pairwise coprime or a singleton: A087087.
The version counting partitions is 1 + A335240.
All of the following pertain to compositions in standard order:
- Length is A000120.
- The parts are row k of A066099.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
Cf. A007360, A048793, A051424, A101268, A272919, A291166, A302569, A326675, A333227, A333228, A335238.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!(Length[stc[#]]==1||CoprimeQ@@stc[#])&]

A337452 Number of relatively prime strict integer partitions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 2, 6, 3, 9, 7, 11, 11, 20, 15, 28, 24, 35, 36, 55, 47, 73, 71, 95, 96, 136, 123, 180, 177, 226, 235, 305, 299, 403, 406, 503, 523, 668, 662, 852, 873, 1052, 1115, 1370, 1391, 1720, 1784, 2125, 2252, 2701, 2786, 3348, 3520, 4116

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

			The a(5) = 1 through a(16) = 11 partitions (A = 10, B = 11, C = 12, D = 13):
  32  43  53  54   73   65   75   76   95    87    97
      52      72   532  74   543  85   B3    B4    B5
              432       83   732  94   653   D2    D3
                        92        A3   743   654   754
                        542       B2   752   753   763
                        632       643  932   762   853
                                  652  5432  843   943
                                  742        852   952
                                  832        942   B32
                                             A32   6532
                                             6432  7432

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

A078374 is the version allowing 1's.
A302698 is the non-strict version.
A332004 is the ordered version allowing 1's.
A337450 is the ordered non-strict version.
A337451 is the ordered version.
A337485 is the pairwise coprime version.
A000837 counts relatively prime partitions.
A078374 counts relatively prime strict partitions.
A002865 counts partitions with no 1's.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
Cf. A007359, A101268, A289509, A337485, A337563.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

A335239 Numbers k such that the k-th composition in standard-order (A066099) does not have all pairwise coprime parts, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 21, 22, 26, 32, 34, 36, 40, 42, 43, 45, 46, 53, 54, 58, 64, 69, 70, 73, 74, 76, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 98, 100, 104, 106, 107, 109, 110, 117, 118, 122, 128, 130, 136, 138, 139, 141, 142, 146, 147, 148, 149, 150, 153

Offset: 1

Gus Wiseman, May 28 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

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 compositions begins:
    0: ()            45: (2,1,2,1)     86: (2,2,1,2)
    2: (2)           46: (2,1,1,2)     87: (2,2,1,1,1)
    4: (3)           53: (1,2,2,1)     88: (2,1,4)
    8: (4)           54: (1,2,1,2)     90: (2,1,2,2)
   10: (2,2)         58: (1,1,2,2)     91: (2,1,2,1,1)
   16: (5)           64: (7)           93: (2,1,1,2,1)
   21: (2,2,1)       69: (4,2,1)       94: (2,1,1,1,2)
   22: (2,1,2)       70: (4,1,2)       98: (1,4,2)
   26: (1,2,2)       73: (3,3,1)      100: (1,3,3)
   32: (6)           74: (3,2,2)      104: (1,2,4)
   34: (4,2)         76: (3,1,3)      106: (1,2,2,2)
   36: (3,3)         81: (2,4,1)      107: (1,2,2,1,1)
   40: (2,4)         82: (2,3,2)      109: (1,2,1,2,1)
   42: (2,2,2)       84: (2,2,3)      110: (1,2,1,1,2)
   43: (2,2,1,1)     85: (2,2,2,1)    117: (1,1,2,2,1)

The complement is A333227.
The version without singletons is A335236.
Ignoring repeated parts gives A335238.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime partitions are counted by A327516.
Non-coprime partitions are counted by A335240.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Number of distinct parts is A334028.
Cf. A007360, A048793, A101268, A291166, A302569, A335235, A335237.

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

A337450 Number of relatively prime compositions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 7, 5, 17, 17, 54, 51, 143, 168, 358, 482, 986, 1313, 2583, 3663, 6698, 9921, 17710, 26489, 46352, 70928, 121137, 188220, 317810, 497322, 832039, 1313501, 2177282, 3459041, 5702808, 9094377, 14930351, 23895672, 39084070, 62721578

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(5) = 2 through a(10) = 17 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)    (3,5)    (2,7)      (3,7)
  (3,2)     (3,4)    (5,3)    (4,5)      (7,3)
            (4,3)    (2,3,3)  (5,4)      (2,3,5)
            (5,2)    (3,2,3)  (7,2)      (2,5,3)
            (2,2,3)  (3,3,2)  (2,2,5)    (3,2,5)
            (2,3,2)           (2,3,4)    (3,3,4)
            (3,2,2)           (2,4,3)    (3,4,3)
                              (2,5,2)    (3,5,2)
                              (3,2,4)    (4,3,3)
                              (3,4,2)    (5,2,3)
                              (4,2,3)    (5,3,2)
                              (4,3,2)    (2,2,3,3)
                              (5,2,2)    (2,3,2,3)
                              (2,2,2,3)  (2,3,3,2)
                              (2,2,3,2)  (3,2,2,3)
                              (2,3,2,2)  (3,2,3,2)
                              (3,2,2,2)  (3,3,2,2)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 171 terms from Fausto A. C. Cariboni)

A000740 is the version allowing 1's.
2*A055684(n) is the case of length 2.
A302697 ranks the unordered case.
A302698 is the unordered version.
A337451 is the strict version.
A337452 is the unordered strict version.
A000837 counts relatively prime partitions.
A002865 counts partitions with no 1's.
A101268 counts singleton or pairwise coprime compositions.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337462 counts pairwise coprime compositions.
Cf. A000010, A007359, A023023, A101268, A178472, A289509, A302568, A337485.

b:= proc(n, g) option remember; `if`(n=0,
     `if`(g=1, 1, 0), add(b(n-j, igcd(g, j)), j=2..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..42);

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

A337451 Number of relatively prime strict compositions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 10, 8, 20, 14, 34, 52, 72, 90, 146, 172, 244, 390, 502, 680, 956, 1218, 1686, 2104, 3436, 4078, 5786, 7200, 10108, 12626, 17346, 20876, 32836, 38686, 53674, 67144, 91528, 113426, 152810, 189124, 245884, 343350, 428494, 552548, 719156

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

			The a(5) = 2 through a(10) = 8 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)    (3,7)
  (3,2)     (3,4)  (5,3)  (4,5)    (7,3)
            (4,3)         (5,4)    (2,3,5)
            (5,2)         (7,2)    (2,5,3)
                          (2,3,4)  (3,2,5)
                          (2,4,3)  (3,5,2)
                          (3,2,4)  (5,2,3)
                          (3,4,2)  (5,3,2)
                          (4,2,3)
                          (4,3,2)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

A032022 does not require relative primality.
A302698 is the unordered non-strict version.
A332004 is the version allowing 1's.
A337450 is the non-strict version.
A337452 is the unordered version.
A000837 counts relatively prime partitions.
A032020 counts strict compositions.
A078374 counts strict relatively prime partitions.
A002865 counts partitions with no 1's.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337462 counts pairwise coprime compositions.
A337561 counts strict pairwise coprime compositions.
Cf. A000010, A007359, A101268, A178472, A216652, A289509, A337562, A337563.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

A101391 Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1,x_2,...,x_k) = 1 (1<=k<=n).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 3, 1, 0, 4, 6, 4, 1, 0, 2, 9, 10, 5, 1, 0, 6, 15, 20, 15, 6, 1, 0, 4, 18, 34, 35, 21, 7, 1, 0, 6, 27, 56, 70, 56, 28, 8, 1, 0, 4, 30, 80, 125, 126, 84, 36, 9, 1, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 0, 4, 42, 154, 325, 461, 462, 330, 165, 55, 11, 1, 0, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1

Offset: 1

Emeric Deutsch, Jan 26 2005

If instead we require that the individual parts (x_i,x_j) be relatively prime, we get A282748. This is the question studied by Shonhiwa (2006). - N. J. A. Sloane, Mar 05 2017.

			T(6,3)=9 because we have 411,141,114 and the six permutations of 123 (222 does not qualify).
T(8,3)=18 because binomial(0,2)*mobius(8/1)+binomial(1,2)*mobius(8/2)+binomial(3,2)*mobius(8/4)+binomial(7,2)*mobius(8/8)=0+0+(-3)+21=18.
Triangle begins:
   1;
   0,  1;
   0,  2,  1;
   0,  2,  3,   1;
   0,  4,  6,   4,   1;
   0,  2,  9,  10,   5,   1;
   0,  6, 15,  20,  15,   6,   1;
   0,  4, 18,  34,  35,  21,   7,   1;
   0,  6, 27,  56,  70,  56,  28,   8,   1;
   0,  4, 30,  80, 125, 126,  84,  36,   9,   1;
   0, 10, 45, 120, 210, 252, 210, 120,  45,  10,  1;
   0,  4, 42, 154, 325, 461, 462, 330, 165,  55, 11,  1;
   0, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1;
  ...
From _Gus Wiseman_, Oct 19 2020: (Start)
Row n = 6 counts the following compositions:
  (15)  (114)  (1113)  (11112)  (111111)
  (51)  (123)  (1122)  (11121)
        (132)  (1131)  (11211)
        (141)  (1212)  (12111)
        (213)  (1221)  (21111)
        (231)  (1311)
        (312)  (2112)
        (321)  (2121)
        (411)  (2211)
               (3111)
Missing are: (42), (24), (33), (222).
(End)

Alois P. Heinz, Rows n = 1..200, flattened
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44 (2006), no. 4, 316-323.

Mirror image of A039911.
Row sums are A000740.
Columns 2-9 are: A000010, A000741, A000742, A000743, A023031, A023032, A023033, A023034.
A000837 counts relatively prime partitions.
A135278 counts compositions by length.
A282748 is the pairwise coprime instead of relatively prime version.
A282750 is the unordered version.
A291166 ranks these compositions (evidently).
T(2n+1,n+1) gives A000984.
Cf. A007359, A008683, A085411, A101268, A289508, A289509, A302697, A332004, A337485, A337563.

with(numtheory): T:=proc(n,k) local d, j, b: d:=divisors(n): for j from 1 to tau(n) do b[j]:=binomial(d[j]-1,k-1)*mobius(n/d[j]) od: sum(b[i],i=1..tau(n)) end: for n from 1 to 14 do seq(T(n,k),k=1..n) od; # yields the sequence in triangular form
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, `if`(g=1, 1, 0),
      expand(add(b(n-j, igcd(g, j))*x, j=1..n)))
    end:
T:= (n, k)-> coeff(b(n,0),x,k):
seq(seq(T(n,k), k=1..n), n=1..14);  # Alois P. Heinz, May 05 2025

t[n_, k_] := Sum[Binomial[d-1, k-1]*MoebiusMu[n/d], {d, Divisors[n]}]; Table[t[n, k], {n, 2, 14}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jan 20 2014 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],GCD@@#==1&]],{n,10},{k,2,n}] (* change {k,2,n} to {k,1,n} for the version with zeros. - Gus Wiseman, Oct 19 2020 *)

T(n, k) = sumdiv(n, d, binomial(d-1, k-1)*moebius(n/d)); \\ Michel Marcus, Mar 09 2016

T(n,k) = Sum_{d|n} binomial(d-1,k-1)*mobius(n/d).

Sum_{k=1..n} k * T(n,k) = A085411(n). - Alois P. Heinz, May 05 2025

Definition clarified by N. J. A. Sloane, Mar 05 2017

Edited by Alois P. Heinz, May 05 2025

A291165 Disconnected Haar graph numbers.

Original entry on oeis.org

2, 4, 8, 10, 16, 32, 34, 36, 40, 42, 64, 128, 130, 136, 138, 160, 162, 168, 170, 256, 260, 288, 292, 512, 514, 520, 522, 528, 544, 546, 552, 554, 640, 642, 648, 650, 672, 674, 680, 682, 1024, 2048, 2050, 2052, 2056, 2058, 2080, 2082, 2084, 2088, 2090, 2176, 2178, 2184

Offset: 1

Eric W. Weisstein, Aug 19 2017

nonn

Includes numbers of the form 2^n.

Complement of A291166.

Eric Weisstein's World of Mathematics, Disconnected Graph
Eric Weisstein's World of Mathematics, Haar Graph

Cf. A291166 (connected).

A326672 The positions of ones in the binary expansion of n have integer geometric mean.

Original entry on oeis.org

1, 2, 4, 8, 9, 13, 16, 18, 26, 32, 36, 52, 64, 72, 104, 128, 144, 208, 256, 257, 288, 321, 416, 512, 514, 576, 642, 832, 1024, 1028, 1152, 1284, 1664, 2048, 2056, 2304, 2568, 3328, 4096, 4112, 4608, 5136, 6656, 8192, 8224, 9216, 10272, 13312, 16384, 16448

Offset: 1

Gus Wiseman, Jul 17 2019

Amiram Eldar, Table of n, a(n) for n = 1..500
Wikipedia, Geometric mean.

Partitions with integer geometric mean are A067539.
Subsets with integer geometric mean are A326027.
Factorizations with integer geometric mean are A326028.
Numbers whose binary expansion positions have integer mean are A326669.
Numbers whose binary expansion positions are relatively prime are A326674.
Numbers whose reversed binary expansion positions have integer geometric mean are A326673.
Cf. A000120, A051293, A070939, A291166, A326625.

Select[Range[100],IntegerQ[GeometricMean[Join@@Position[IntegerDigits[#,2],1]]]&]

A059519 Number of partitions of n all of whose subpartitions sum to distinct values. Partition(n) = [a, b, c...] where 2n = 2^a + 2^b + 2^c + ...

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 24, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 44, 48, 50, 52, 56, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 80, 81, 84, 88, 96, 98, 100, 104, 112, 116, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 139, 140

Offset: 1

Marc LeBrun, Jan 19 2001

Partition encoding as in A029931. Complement of A059520.
From Gus Wiseman, Jul 22 2019: (Start)
These are numbers whose positions of 1's in their reversed binary expansion form a strict knapsack partition (A275972). The initial terms together with their corresponding partitions are:
(1)
(2)
(2,1)
(3)
(3,1)
(3,2)
(4)
(4,1)
(4,2)
(4,2,1)
(4,3)
(4,3,2)
(5)
(5,1)
(5,2)
(5,2,1)
(5,3)
(End)

			14=2+4+8 so Partition(14) = [2,3,4], whose sub-sums are 0,2,3,4,5,6,7 and 14.

Cf. A000120, A029931, A048793, A059520, A070939, A108917, A272020, A275972, A299702, A326015.

Other sequences classifying numbers by their binary indices: A291166 (relatively prime), A295235 (arithmetic progression), A326669 (integer average), A326675 (pairwise coprime).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],UnsameQ@@Total/@Subsets[bpe[#]]&] (* Gus Wiseman, Jul 22 2019 *)

cp's OEIS Frontend

A359495 Sum of positions of 1's in binary expansion minus sum of positions of 1's in reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

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A335236 Numbers k such that the k-th composition in standard order (A066099) is not a singleton nor pairwise coprime.

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A337452 Number of relatively prime strict integer partitions of n with no 1's.

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A335239 Numbers k such that the k-th composition in standard-order (A066099) does not have all pairwise coprime parts, where a singleton is not coprime unless it is (1).

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A337450 Number of relatively prime compositions of n with no 1's.

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A337451 Number of relatively prime strict compositions of n with no 1's.

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A101391 Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1,x_2,...,x_k) = 1 (1<=k<=n).

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A291165 Disconnected Haar graph numbers.

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