A291166 -id:A291166 - cp's OEIS frontend

A328671 Numbers whose binary indices are relatively prime and pairwise indivisible.

1, 6, 12, 18, 20, 22, 24, 28, 48, 56, 66, 68, 70, 72, 76, 80, 82, 84, 86, 88, 92, 96, 104, 112, 120, 132, 144, 148, 176, 192, 196, 208, 212, 224, 240, 258, 264, 272, 274, 280, 296, 304, 312, 320, 322, 328, 336, 338, 344, 352, 360, 368, 376, 384, 400, 416, 432

Offset: 1

Gus Wiseman, Oct 29 2019

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 sequence of terms together with their binary expansions and binary indices begins:
    1:         1 ~ {1}
    6:       110 ~ {2,3}
   12:      1100 ~ {3,4}
   18:     10010 ~ {2,5}
   20:     10100 ~ {3,5}
   22:     10110 ~ {2,3,5}
   24:     11000 ~ {4,5}
   28:     11100 ~ {3,4,5}
   48:    110000 ~ {5,6}
   56:    111000 ~ {4,5,6}
   66:   1000010 ~ {2,7}
   68:   1000100 ~ {3,7}
   70:   1000110 ~ {2,3,7}
   72:   1001000 ~ {4,7}
   76:   1001100 ~ {3,4,7}
   80:   1010000 ~ {5,7}
   82:   1010010 ~ {2,5,7}
   84:   1010100 ~ {3,5,7}
   86:   1010110 ~ {2,3,5,7}
   88:   1011000 ~ {4,5,7}

The version for prime indices (instead of binary indices) is A328677.
Numbers whose binary indices are relatively prime are A291166.
Numbers whose distinct prime indices are pairwise indivisible are A316476.
BII-numbers of antichains are A326704.
Relatively prime partitions whose distinct parts are pairwise indivisible are A328676, with strict case A328678.
Cf. A000120, A121016, A285573, A303362, A304713, A305148, A316468, A316475.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],GCD@@bpe[#]==1&&stableQ[bpe[#],Divisible]&]

Intersection of A291166 with A326704.

A359401 Nonnegative integers whose sum of positions of 1's in their binary expansion is greater than the sum of positions of 1's in their reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

Original entry on oeis.org

11, 19, 23, 35, 37, 39, 43, 47, 55, 67, 69, 71, 75, 77, 79, 83, 87, 91, 95, 103, 111, 131, 133, 134, 135, 137, 139, 141, 142, 143, 147, 149, 151, 155, 157, 158, 159, 163, 167, 171, 173, 175, 179, 183, 187, 191, 199, 203, 207, 215, 223, 239, 259, 261, 262, 263

Offset: 1

Gus Wiseman, Jan 05 2023

First differs from A161601 in having 134, with binary expansion (1,0,0,0,0,1,1,0), positions of 1's 1 + 6 + 7 = 14, reversed 2 + 3 + 8 = 13.

Indices of positive terms in A359495; 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.
A326669 lists numbers with integer mean position of a 1 in binary expansion.
Cf. A000120, A048793, A051293, A053632, A222955, A231204, A291166, A304818, A326672, A326673, A359043.

sap[q_]:=Sum[q[[i]]*(2i-Length[q]-1),{i,Length[q]}];
Select[Range[0,100],sap[IntegerDigits[#,2]]>0&]

A230877(a(n)) > A029931(a(n)).

A326699 Numerator of the average position of a 1 in the reversed binary expansion of n.

Original entry on oeis.org

1, 2, 3, 3, 2, 5, 2, 4, 5, 3, 7, 7, 8, 3, 5, 5, 3, 7, 8, 4, 3, 10, 11, 9, 10, 11, 3, 4, 13, 7, 3, 6, 7, 4, 3, 9, 10, 11, 3, 5, 11, 4, 13, 13, 7, 15, 16, 11, 4, 13, 7, 14, 15, 4, 17, 5, 4, 17, 18, 9, 19, 4, 7, 7, 4, 9, 10, 5, 11, 4, 13, 11, 4, 13, 7, 14, 15, 4

Offset: 1

Gus Wiseman, Jul 20 2019

			The sequence of fractions begins: 1, 2, 3/2, 3, 2, 5/2, 2, 4, 5/2, 3, 7/3, 7/2, 8/3, 3, 5/2, 5, 3, 7/2, 8/3, 4.
For example, the reversed binary expansion of 18 is (0,1,0,0,1), and the average of {2,5} is 7/2, so a(18) = 7.

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

Cf. A000120, A029931, A035327, A048793, A070939, A291166, A295235, A326669, A326674, A326700 (denominators).

f:= proc(n) local L;
  L:= convert(n,base,2);
  L:= select(t -> L[t]=1, [$1..nops(L)]);
  numer(convert(L,`+`)/nops(L))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 07 2019

Table[Numerator[Mean[Join@@Position[Reverse[IntegerDigits[n,2]],1]]],{n,100}]

A326700 Denominator of the average position of a 1 in the reversed binary expansion of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 20 2019

The sequence of fractions begins: 1, 2, 3/2, 3, 2, 5/2, 2, 4, 5/2, 3, 7/3, 7/2, 8/3, 3, 5/2, 5, 3, 7/2, 8/3, 4.
For example, the reversed binary expansion of 18 is (0,1,0,0,1), and the average of {2,5} is 7/2, so a(18) = 2.
a(n) divides A000120(n). - Robert Israel, Oct 07 2019

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

Positions of 1's are A326669.

Cf. A000120, A029931, A035327, A048793, A070939, A291166, A295235, A326674, A326699 (numerators).

f:= proc(n) local L;
  L:= convert(n,base,2);
  L:= select(t -> L[t]=1, [$1..nops(L)]);
  denom(convert(L,`+`)/nops(L))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 07 2019

Table[Denominator[Mean[Join@@Position[Reverse[IntegerDigits[n,2]],1]]],{n,100}]

A328672 Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 4, 1, 1, 2, 7, 1, 6, 1, 3, 3, 10, 1, 9, 3, 5, 4, 17, 1, 23, 6, 7, 6, 20, 3, 36, 9, 15, 7, 45, 5, 56, 14, 17, 20, 65, 7, 83, 18, 40

Offset: 0

Gus Wiseman, Oct 29 2019

nonn

Positions of terms greater than 1 are {31, 37, 41, 43, 46, 47, 49, ...}.

A partition with no two distinct parts relatively prime is said to be intersecting.

			Examples:
  a(31) = 2:         a(46) = 2:
    (15,10,6)          (15,15,10,6)
    (1^31)             (1^46)
  a(37) = 3:         a(47) = 7:
    (15,12,10)         (20,15,12)
    (15,10,6,6)        (21,14,12)
    (1^37)             (20,15,6,6)
  a(41) = 4:           (21,14,6,6)
    (20,15,6)          (15,12,10,10)
    (21,14,6)          (15,10,10,6,6)
    (15,10,10,6)       (1^47)
    (1^41)           a(49) = 6:
  a(43) = 4:           (24,15,10)
    (18,15,10)         (18,15,10,6)
    (15,12,10,6)       (15,12,12,10)
    (15,10,6,6,6)      (15,12,10,6,6)
    (1^43)             (15,10,6,6,6,6)
                       (1^39)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400

The Heinz numbers of these partitions are A328679.
The strict case is A318715.
The version for non-isomorphic multiset partitions is A319759.
Relatively prime partitions are A000837.
Intersecting partitions are A328673.
Cf. A078374, A285573, A289509, A291166, A303140, A305148, A316476, A326910, A326912.

Table[Length[Select[IntegerPartitions[n],GCD@@#==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]]&]],{n,0,32}]

a(n > 0) = A202425(n) + 1.

A335237 Numbers whose binary indices are not a singleton nor pairwise coprime.

Original entry on oeis.org

0, 10, 11, 14, 15, 26, 27, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 74, 75, 78, 79, 90, 91, 94, 95, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 114, 115, 116

Offset: 1

Gus Wiseman, May 28 2020

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 sequence of terms together with their binary expansions and binary indices begins:
    0:       0 ~ {}
   10:    1010 ~ {2,4}
   11:    1011 ~ {1,2,4}
   14:    1110 ~ {2,3,4}
   15:    1111 ~ {1,2,3,4}
   26:   11010 ~ {2,4,5}
   27:   11011 ~ {1,2,4,5}
   30:   11110 ~ {2,3,4,5}
   31:   11111 ~ {1,2,3,4,5}
   34:  100010 ~ {2,6}
   35:  100011 ~ {1,2,6}
   36:  100100 ~ {3,6}
   37:  100101 ~ {1,3,6}
   38:  100110 ~ {2,3,6}
   39:  100111 ~ {1,2,3,6}
   40:  101000 ~ {4,6}
   41:  101001 ~ {1,4,6}
   42:  101010 ~ {2,4,6}
   43:  101011 ~ {1,2,4,6}
   44:  101100 ~ {3,4,6}

The version for prime indices is A316438.
The version for standard compositions is A335236.
Numbers whose binary indices are pairwise coprime or a singleton: A087087.
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.
Cf. A007360, A048793, A051424, A101268, A291166, A302569, A326675, A333227, A333228, A335235, A335239.

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

Complement in A001477 of A326675 and A000079.

A333941 Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 2, 3, 0, 0, 2, 4, 6, 4, 0, 0, 4, 6, 9, 8, 5, 0, 0, 2, 6, 15, 20, 15, 6, 0, 0, 4, 8, 24, 32, 35, 18, 7, 0, 0, 3, 10, 27, 56, 70, 54, 28, 8, 0, 0, 4, 12, 42, 84, 125, 120, 84, 32, 9, 0, 0, 2, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset: 0

Gus Wiseman, Apr 16 2020

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

			Triangle begins:
   1
   0   1
   0   2   0
   0   2   2   0
   0   3   2   3   0
   0   2   4   6   4   0
   0   4   6   9   8   5   0
   0   2   6  15  20  15   6   0
   0   4   8  24  32  35  18   7   0
   0   3  10  27  56  70  54  28   8   0
   0   4  12  42  84 125 120  84  32   9   0
   0   2  10  45 120 210 252 210 120  45  10   0
   0   6  18  66 168 335 450 462 320 162  50  11   0
Row n = 6 counts the following compositions (empty columns indicated by dots):
  .  (6)       (15)    (114)  (1113)  (11112)  .
     (33)      (24)    (123)  (1122)  (11121)
     (222)     (42)    (132)  (1131)  (11211)
     (111111)  (51)    (141)  (1221)  (12111)
               (1212)  (213)  (1311)  (21111)
               (2121)  (231)  (2112)
                       (312)  (2211)
                       (321)  (3111)
                       (411)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Column k = 1 is A000005.
Row sums are A011782.
Diagonal T(2n,n) is A045630(n).
The strict version is A072574.
A version counting runs is A238279.
Column k = n - 1 is A254667.
Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
Period of binary expansion is A302291.
Numbers whose prime signature is aperiodic are A329139.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Rotational symmetries are counted by A138904.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Reversed necklaces are A333943.
Cf. A000031, A001037, A008965, A019536, A211100, A291166, A328595, A328596, A329312, A329313, A329326.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Function[c,Length[Union[Array[RotateRight[c,#]&,Length[c]]]]==k]]],{n,0,10},{k,0,n}]

T(n,k)=if(n==0, k==0, sumdiv(n, m, sumdiv(gcd(k,m), d, moebius(d)*binomial(m/d-1, k/d-1)))) \\ Andrew Howroyd, Jan 19 2023

T(n,k) = Sum_{m|n} Sum_{d|gcd(k,m)} mu(d)*binomial(m/d-1, k/d-1) for n > 0. - Andrew Howroyd, Jan 19 2023

A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

Original entry on oeis.org

17719, 40807, 43381, 50431, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239, 215267, 219271, 230347, 246703, 249587, 249899, 279371, 286897, 289007, 296993, 300847

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   17719: {6,10,15}
   40807: {6,14,21}
   43381: {6,15,20}
   50431: {10,12,15}
   74269: {6,10,45}
   83143: {10,15,18}
  101543: {6,21,28}
  105703: {6,15,40}
  116143: {12,14,21}
  121307: {10,15,24}
  123469: {12,15,20}
  139919: {6,15,50}
  140699: {6,22,33}
  142883: {6,10,75}
  171613: {6,14,63}
  181831: {6,20,45}
  185803: {10,14,35}
  191479: {14,18,21}
  203557: {15,18,20}
  205813: {10,15,36}
  211381: {10,12,45}
  213239: {6,15,70}
  215267: {6,10,105}
  219271: {6,26,39}
  230347: {6,6,10,15}

These are the Heinz numbers of the partitions counted by A202425.
Terms of A328679 that are not powers of 2.
The strict case is A318716 (preceded by 2).
A ranking using binary indices (instead of prime indices) is A326912.
Heinz numbers of relatively prime partitions are A289509.
Cf. A000837, A056239, A112798, A200976, A291166, A302796, A316476, A318715, A319752, A319759, A328336, A328672, A328677, A328867.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dv=Select[Range[100000],GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Tuples[Union[primeMS[#]],2]]&]

A357000 Number of non-isomorphic cyclic Haar graphs on 2*n nodes.

Original entry on oeis.org

1, 2, 3, 5, 5, 12, 9, 22, 21, 44, 29, 157, 73, 244, 367, 649, 521, 2624, 1609, 7385, 8867, 19400, 16769, 92529, 67553, 216274, 277191, 815557, 662369, 4500266, 2311469

Offset: 1

Pontus von Brömssen, Sep 08 2022

The first value of n for which a(n) < A002729(n) - 1 is n = 8. This is because the first counterexample to the bicirculant analog to Ádám's conjecture occurs for n = 8. In the terminology of Hladnik, Marušič, and Pisanski, the smallest integer pair (i,j) such that i and j are Haar equivalent (i.e., the cyclic Haar graphs with indices i and j are isomorphic) but not cyclically equivalent (see A357005) is (141,147). See also A357001 and A357002.

Terms a(1)-a(29) were found by generating the cyclic Haar graphs with indices in A333764, and filtering out isomorphic graphs using Brendan McKay's software nauty.

Milan Hladnik, Dragan Marušič, and Tomaž Pisanski, Cyclic Haar graphs, Discrete Mathematics 244 (2002), 137-152.
Eric Weisstein's World of Mathematics, Haar Graph

Cf. A002729, A008965, A049287, A091696, A137706, A291166, A333764, A339502, A357001, A357002, A357003, A357004, A357005, A357006.

a(n) is the number of terms k of A137706 in the interval 2^(n-1) <= k < 2^n.
a(n) is the number of fixed points k of A357004 in the interval 2^(n-1) <= k < 2^n.
a(n) <= A002729(n)-1 <= A091696(n) <= A008965(n).

a(30) from Eric W. Weisstein, Jun 27 2023

a(31) from Eric W. Weisstein, Jun 28 2023

A328679 Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the parts.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 17719, 32768, 40807, 43381, 50431, 65536, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 131072, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

Equals the union A000079 and A328868.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A partition with no two distinct parts relatively prime is said to be intersecting.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      8: {1,1,1}
     16: {1,1,1,1}
     32: {1,1,1,1,1}
     64: {1,1,1,1,1,1}
    128: {1,1,1,1,1,1,1}
    256: {1,1,1,1,1,1,1,1}
    512: {1,1,1,1,1,1,1,1,1}
   1024: {1,1,1,1,1,1,1,1,1,1}
   2048: {1,1,1,1,1,1,1,1,1,1,1}
   4096: {1,1,1,1,1,1,1,1,1,1,1,1}
   8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
  16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  17719: {6,10,15}
  32768: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  40807: {6,14,21}
  43381: {6,15,20}
  50431: {10,12,15}
  65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

These are the Heinz numbers of the partitions counted by A328672.
Terms that are not powers of 2 are A328868.
The strict case is A318716.
The version without global relative primality is A328867.
A ranking using binary indices (instead of prime indices) is A326912.
The version for non-isomorphic multiset partitions is A319759.
The version for divisibility (instead of relative primality) is A328677.
Heinz numbers of relatively prime partitions are A289509.
Cf. A000837, A056239, A112798, A200976, A202425, A289509, A291166, A302796, A316476, A318715, A319752, A328336.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],#==1||GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[primeMS[#]],{2}]]&]

cp's OEIS Frontend

A328671 Numbers whose binary indices are relatively prime and pairwise indivisible.

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A359401 Nonnegative integers whose sum of positions of 1's in their binary expansion is greater than the sum of positions of 1's in their reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

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A326699 Numerator of the average position of a 1 in the reversed binary expansion of n.

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A326700 Denominator of the average position of a 1 in the reversed binary expansion of n.

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A328672 Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.

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A335237 Numbers whose binary indices are not a singleton nor pairwise coprime.

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A333941 Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.

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A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

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