A000740 -id:A000740 - cp's OEIS frontend

A328594 Numbers whose binary expansion is aperiodic.

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Oct 22 2019

nonn

A finite sequence is aperiodic if all of its cyclic rotations are distinct. See A000740 or A027375 for details.

Also numbers k such that the k-th composition in standard order is aperiodic. 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. - Gus Wiseman, Apr 28 2020

			The sequence of terms together with their binary expansions and binary indices begins:
   0:     0 ~ {}
   1:     1 ~ {1}
   2:    10 ~ {2}
   4:   100 ~ {3}
   5:   101 ~ {1,3}
   6:   110 ~ {2,3}
   8:  1000 ~ {4}
   9:  1001 ~ {1,4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  13:  1101 ~ {1,3,4}
  14:  1110 ~ {2,3,4}
  16: 10000 ~ {5}
  17: 10001 ~ {1,5}
  18: 10010 ~ {2,5}
  19: 10011 ~ {1,2,5}
  20: 10100 ~ {3,5}
  21: 10101 ~ {1,3,5}
  22: 10110 ~ {2,3,5}
  23: 10111 ~ {1,2,3,5}
  24: 11000 ~ {4,5}

The complement is A121016.
The version for prime indices is A085971.
Numbers without proper integer roots are A007916.
Necklaces are A328595.
Lyndon words are A328596.
Aperiodic compositions are A000740.
Aperiodic binary sequences are A027375.
Cf. A000120, A000939, A014081, A065609, A069010, A275692, A323867, A334030.

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Select[Range[0,100],aperQ[IntegerDigits[#,2]]&]

A329744 Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 1, 6, 6, 2, 1, 3, 15, 9, 4, 0, 1, 1, 22, 22, 16, 2, 0, 1, 3, 41, 38, 37, 8, 0, 0, 1, 2, 72, 69, 86, 26, 0, 0, 0, 1, 3, 129, 124, 175, 78, 2, 0, 0, 0, 1, 1, 213, 226, 367, 202, 14, 0, 0, 0, 0, 1, 5, 395, 376, 750, 469, 52, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 21 2019

A composition of n is a finite sequence of positive integers with sum n.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Triangle begins:
   1
   1   1
   1   1   2
   1   2   3   2
   1   1   6   6   2
   1   3  15   9   4   0
   1   1  22  22  16   2   0
   1   3  41  38  37   8   0   0
   1   2  72  69  86  26   0   0   0
   1   3 129 124 175  78   2   0   0   0
   1   1 213 226 367 202  14   0   0   0   0
   1   5 395 376 750 469  52   0   0   0   0   0
Row n = 6 counts the following compositions:
  (6)  (33)      (15)    (114)    (1131)
       (222)     (24)    (411)    (1311)
       (111111)  (42)    (1113)   (11121)
                 (51)    (1221)   (12111)
                 (123)   (2112)
                 (132)   (3111)
                 (141)   (11112)
                 (213)   (11211)
                 (231)   (21111)
                 (312)
                 (321)
                 (1122)
                 (1212)
                 (2121)
                 (2211)

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Row sums are A000079.
Column k = 1 is A032741.
Column k = 2 is A329745.
Column k = n - 2 is A329743.
The version for partitions is A329746.
The version with rows reversed is A329750.
Cf. A000740, A008965, A098504, A242882, A318928, A329747.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==k&]],{n,10},{k,0,n-1}]

A325545 Number of compositions of n with distinct differences.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 17, 34, 59, 105, 166, 279, 442, 730, 1157, 1927, 3045, 4741, 7527, 11667, 18048, 27928, 43334, 65861, 101385, 153404, 232287, 347643, 523721, 780083, 1165331, 1725966, 2561625, 3773838, 5561577, 8151209, 11920717, 17364461, 25269939, 36635775

Offset: 0

Gus Wiseman, May 10 2019

nonn

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

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)   (3)   (4)    (5)     (6)
       (11)  (12)  (13)   (14)    (15)
             (21)  (22)   (23)    (24)
                   (31)   (32)    (33)
                   (112)  (41)    (42)
                   (121)  (113)   (51)
                   (211)  (122)   (114)
                          (131)   (132)
                          (212)   (141)
                          (221)   (213)
                          (311)   (231)
                          (1121)  (312)
                          (1211)  (411)
                                  (1131)
                                  (1221)
                                  (1311)
                                  (2112)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..70
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A059966, A070211, A175342, A242882, A325325, A325352, A325546, A325547, A325548, A325551, A325554.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Differences[#]&]],{n,0,15}]

More terms from Alois P. Heinz, May 11 2019

A101268 Number of compositions of n into pairwise relatively prime parts.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 22, 38, 63, 101, 160, 254, 403, 635, 984, 1492, 2225, 3281, 4814, 7044, 10271, 14889, 21416, 30586, 43401, 61205, 85748, 119296, 164835, 226423, 309664, 422302, 574827, 781237, 1060182, 1436368, 1942589, 2622079, 3531152, 4742316, 6348411

Offset: 0

Vladeta Jovovic, Dec 18 2004

nonn

Here a singleton is always considered pairwise relatively prime. Compare to A337462. - Gus Wiseman, Oct 18 2020

			From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(5) = 13 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (31)    (23)
             (111)  (112)   (32)
                    (121)   (41)
                    (211)   (113)
                    (1111)  (131)
                            (311)
                            (1112)
                            (1121)
                            (1211)
                            (2111)
                            (11111)
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500 (terms 0..400 from Alois P. Heinz)
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44 (2006), no. 4, 316-323.

Row sums of A282748.
A051424 is the unordered version, with strict case A007360.
A335235 ranks these compositions.
A337461 counts these compositions of length 3, with unordered version A307719 and unordered strict version A220377.
A337462 does not consider a singleton to be coprime unless it is (1), with strict version A337561.
A337562 is the strict case.
A337664 looks only at distinct parts, with non-constant version A337665.
A000740 counts relatively prime compositions, with strict case A332004.
A178472 counts compositions with a common factor.
Cf. A087087, A302569, A305713, A326675, A327516, A328673, A333227, A333228.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]<=1||CoprimeQ@@#&]],{n,0,10}] (* Gus Wiseman, Oct 18 2020 *)

It seems that no formula is known.

a(0)=1 prepended by Alois P. Heinz, Jun 14 2017

A303386 Number of aperiodic factorizations of n > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 2, 4, 1, 5, 1, 6, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 7, 1, 5, 1, 7, 5

Offset: 2

Gus Wiseman, Apr 23 2018

nonn

An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.

			The a(36) = 7 aperiodic factorizations are (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), and (36). Missing from this list are (2*2*3*3) and (6*6).

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A000740, a(2^n) = A000837(n), A001055, A007716, A007916, A045778, A052409, A052410, A100953, A162247, A275024, A275870, A281113, A281116, A301700, A302242, A303431.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],GCD@@Length/@Split[#]===1&]],{n,2,100}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A303386(n) = if(1==n,n,my(r); sumdiv(A052409(n),d, ispower(n,d,&r); moebius(d)*A001055(r))); \\ Antti Karttunen, Sep 25 2018

a(n) = Sum_{d|A052409(n)} mu(d) * A001055(n^(1/d)), where mu = A008683.

More terms from Antti Karttunen, Sep 25 2018

A351013 Number of integer compositions of n with all distinct runs.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 48, 88, 161, 294, 512, 970, 1634, 2954, 5156, 9119, 15618, 27354, 46674, 80130, 138078, 232286, 394966, 665552, 1123231, 1869714, 3146410, 5186556, 8620936, 14324366, 23529274, 38564554, 63246744, 103578914, 167860584, 274465845

Offset: 0

Gus Wiseman, Feb 09 2022

nonn

			The a(1) = 1 through a(5) = 14 compositions:
  (1)  (2)    (3)      (4)        (5)
       (1,1)  (1,2)    (1,3)      (1,4)
              (2,1)    (2,2)      (2,3)
              (1,1,1)  (3,1)      (3,2)
                       (1,1,2)    (4,1)
                       (2,1,1)    (1,1,3)
                       (1,1,1,1)  (1,2,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)
For example, the composition c = (3,1,1,1,1,2,1,1,3,4,1,1) has runs (3), (1,1,1,1), (2), (1,1), (3), (4), (1,1), and since (3) and (1,1) both appear twice, c is not counted under a(20).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A329739, normal A329740.
These compositions are ranked by A351290, complement A351291.
A000005 counts constant compositions, ranked by A272919.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A059966 counts binary Lyndon compositions, necklaces A008965, aperiodic A000740.
A116608 counts compositions by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329744 counts compositions by runs-resistance.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A025047, A044813, A098504, A098859, A106356, A329738, A328592, A334028, A351015, A351201, A351204.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Split[#]&]],{n,0,10}]

\\ here LahI is A111596 as row polynomials.
LahI(n,y) = {sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
S(n) = {my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p,i,y)*LahI(i,y))}
seq(n)={my(q=S(n)); [subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, subst(q + O(x*x^(n\k)), x, x^k)))]} \\ Andrew Howroyd, Feb 12 2022

Terms a(26) and beyond from Andrew Howroyd, Feb 12 2022

A328595 Numbers whose reversed binary expansion is a necklace.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 40, 42, 44, 48, 52, 54, 56, 58, 60, 62, 63, 64, 72, 80, 84, 88, 92, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 127, 128, 136, 144, 152, 160, 164, 168, 170, 172, 176, 180

Offset: 1

Gus Wiseman, Oct 22 2019

A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their binary expansions and binary indices begins:
   1:      1 ~ {1}
   2:     10 ~ {2}
   3:     11 ~ {1,2}
   4:    100 ~ {3}
   6:    110 ~ {2,3}
   7:    111 ~ {1,2,3}
   8:   1000 ~ {4}
  10:   1010 ~ {2,4}
  12:   1100 ~ {3,4}
  14:   1110 ~ {2,3,4}
  15:   1111 ~ {1,2,3,4}
  16:  10000 ~ {5}
  20:  10100 ~ {3,5}
  24:  11000 ~ {4,5}
  26:  11010 ~ {2,4,5}
  28:  11100 ~ {3,4,5}
  30:  11110 ~ {2,3,4,5}
  31:  11111 ~ {1,2,3,4,5}
  32: 100000 ~ {6}
  36: 100100 ~ {3,6}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

A similar concept is A065609.
The version with the most significant digit ignored is A328607.
Lyndon words are A328596.
Aperiodic words are A328594.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000120, A000740, A001037, A032153, A059966, A275692, A328668.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[100],neckQ[Reverse[IntegerDigits[#,2]]]&]

from itertools import count, islice
from sympy.utilities.iterables import necklaces
def a_gen():
    for n in count(1):
        t = []
        for i in necklaces(n,2):
            if sum(i)>0:
                t.append(sum(2**j for j in range(len(i)) if i[j] > 0))
        yield from sorted(t)
A328595_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, May 24 2024

A301700 Number of aperiodic rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 21, 52, 120, 290, 697, 1713, 4200, 10446, 26053, 65473, 165257, 419357, 1068239, 2732509, 7013242, 18059960, 46641983, 120790324, 313593621, 816046050, 2128101601, 5560829666, 14557746453, 38177226541, 100281484375, 263815322761, 695027102020

Offset: 1

Gus Wiseman, Apr 23 2018

nonn

An unlabeled rooted tree is aperiodic if the multiset of branches of the root is an aperiodic multiset, meaning it has relatively prime multiplicities, and each branch is also aperiodic.

			The a(6) = 10 aperiodic trees are (((((o))))), (((o(o)))), ((o((o)))), ((oo(o))), (o(((o)))), (o(o(o))), ((o)((o))), (oo((o))), (o(o)(o)), (ooo(o)).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000081, A000740, A000837, A001678, A003238, A004111, A007716, A007916, A100953, A276625, A284639, A290689, A298422, A303386, A303431.

arut[n_]:=arut[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[arut/@c]],GCD@@Length/@Split[#]===1&]]/@IntegerPartitions[n-1]];
Table[Length[arut[n]],{n,20}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
MoebiusT(v)={vector(#v, n, sumdiv(n,d,moebius(n/d)*v[d]))}
seq(n)={my(v=[1]); for(n=2, n, v=concat([1], MoebiusT(EulerT(v)))); v} \\ Andrew Howroyd, Sep 01 2018

Terms a(21) and beyond from Andrew Howroyd, Sep 01 2018

A178472 Number of compositions (ordered partitions) of n where the gcd of the part sizes is not 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 8, 4, 17, 1, 38, 1, 65, 19, 128, 1, 284, 1, 518, 67, 1025, 1, 2168, 16, 4097, 256, 8198, 1, 16907, 1, 32768, 1027, 65537, 79, 133088, 1, 262145, 4099, 524408, 1, 1056731, 1, 2097158, 16636, 4194305, 1, 8421248, 64, 16777712, 65539

Offset: 1

Franklin T. Adams-Watters, May 28 2010

nonn

Of course, all part sizes must be greater than 1; that condition alone gives the Fibonacci numbers, which is thus an upper bound.

Also the number of periodic compositions of n, where a sequence is periodic if its cyclic rotations are not all different. Also compositions with non-relatively prime run-lengths. - Gus Wiseman, Nov 10 2019

			For n=6, we have 5 compositions: <6>, <4,2>, <2,4>, <2,2,2>, and <3,3>.
From _Gus Wiseman_, Nov 10 2019: (Start)
The a(2) = 1 through a(9) = 4 non-relatively prime compositions:
  (2)  (3)  (4)    (5)  (6)      (7)  (8)        (9)
            (2,2)       (2,4)         (2,6)      (3,6)
                        (3,3)         (4,4)      (6,3)
                        (4,2)         (6,2)      (3,3,3)
                        (2,2,2)       (2,2,4)
                                      (2,4,2)
                                      (4,2,2)
                                      (2,2,2,2)
The a(2) = 1 through a(9) = 4 periodic compositions:
  11  111  22    11111  33      1111111  44        333
           1111         222              1313      121212
                        1212             2222      212121
                        2121             3131      111111111
                        111111           112112
                                         121121
                                         211211
                                         11111111
The a(2) = 1 through a(9) = 4 compositions with non-relatively prime run-lengths:
  11  111  22    11111  33      1111111  44        333
           1111         222              1133      111222
                        1122             2222      222111
                        2211             3311      111111111
                        111111           111122
                                         112211
                                         221111
                                         11111111
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Hunki Baek, Sejeong Bang, Dongseok Kim, and Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014. See Table 2.

Cf. A000045, A008683, A011782, A178470.
Periodic binary words are A152061.
Cf. A000740, A027375, A059966, A121016, A329140, A329145.

A178472 := n -> (2^n - add(mobius(n/d)*2^d, d in divisors(n)))/2:
seq(A178472(n), n=1..51); # Peter Luschny, Jan 21 2018

Table[2^(n - 1) - DivisorSum[n, MoebiusMu[n/#]*2^(# - 1) &], {n, 51}] (* Michael De Vlieger, Jan 20 2018 *)

vector(60,n,2^(n-1)-sumdiv(n,d,2^(d-1)*moebius(n/d)))

from sympy import mobius, divisors
def A178472(n): return -sum(mobius(n//d)<Chai Wah Wu, Sep 21 2024

a(n) = Sum_{d|n & d

a(n) = 2^(n-1) - A000740(n).

a(n) = A152061(n)/2. - George Beck, Jan 20 2018

a(p) = 1 for p prime. - Chai Wah Wu, Sep 21 2024

Ambiguous term a(0) removed by Max Alekseyev, Jan 02 2012

A303431 Aperiodic tree numbers. Matula-Goebel numbers of aperiodic rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 12, 13, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 37, 39, 40, 41, 44, 45, 47, 48, 50, 52, 54, 55, 58, 60, 61, 62, 65, 66, 71, 72, 74, 75, 78, 79, 80, 82, 87, 88, 89, 90, 93, 94, 96, 99, 101, 104, 108, 109, 110, 111, 113, 116, 117, 120, 122

Offset: 1

Gus Wiseman, Apr 23 2018

nonn

A positive integer is an aperiodic tree number iff either it is equal to 1 or it belongs to A007916 (numbers that are not perfect powers, or numbers whose prime multiplicities are relatively prime) and all of its prime indices are also aperiodic tree numbers, where a prime index of n is a number m such that prime(m) divides n.

			Sequence of aperiodic rooted trees begins:
01 o
02 (o)
03 ((o))
05 (((o)))
06 (o(o))
10 (o((o)))
11 ((((o))))
12 (oo(o))
13 ((o(o)))
15 ((o)((o)))
18 (o(o)(o))
20 (oo((o)))
22 (o(((o))))
24 (ooo(o))
26 (o(o(o)))
29 ((o((o))))
30 (o(o)((o)))
31 (((((o)))))
33 ((o)(((o))))

Cf. A000081, A000740, A000837, A007097, A007916, A052409, A052410, A061775, A111299, A214577, A275024, A276625, A290760, A290822, A291442, A298533, A298536, A301700, A302242, A303386.

zapQ[1]:=True;zapQ[n_]:=And[GCD@@FactorInteger[n][[All,2]]===1,And@@zapQ/@PrimePi/@FactorInteger[n][[All,1]]];
Select[Range[100],zapQ]

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A328594 Numbers whose binary expansion is aperiodic.

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A329744 Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

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A325545 Number of compositions of n with distinct differences.

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A101268 Number of compositions of n into pairwise relatively prime parts.

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A303386 Number of aperiodic factorizations of n > 1.

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PARI

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A351013 Number of integer compositions of n with all distinct runs.

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PARI

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A328595 Numbers whose reversed binary expansion is a necklace.

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Python

A301700 Number of aperiodic rooted trees with n nodes.

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