A045690 -id:A045690 - cp's OEIS frontend

A342085 Number of decreasing chains of distinct superior divisors starting with n.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 6, 1, 5, 1, 4, 2, 2, 1, 11, 2, 2, 3, 4, 1, 7, 1, 10, 2, 2, 2, 15, 1, 2, 2, 10, 1, 6, 1, 4, 5, 2, 1, 26, 2, 5, 2, 4, 1, 11, 2, 10, 2, 2, 1, 21, 1, 2, 5, 20, 2, 6, 1, 4, 2, 7, 1, 39, 1, 2, 5, 4, 2, 6, 1, 23, 6, 2, 1

Offset: 1

Gus Wiseman, Feb 28 2021

nonn

We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.
These chains have first-quotients (in analogy with first-differences) that are term-wise less than or equal to their decapitation (maximum element removed). Equivalently, x <= y^2 for all adjacent x, y. For example, the divisor chain q = 24/8/4/2 has first-quotients (3,2,2), which are less than or equal to (8,4,2), so q is counted under a(24).
Also the number of ordered factorizations of n where each factor is less than or equal to the product of all previous factors.

			The a(n) chains for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:
  2  4    8      12      16        20       24         30       32
     4/2  8/4    12/4    16/4      20/5     24/6       30/6     32/8
          8/4/2  12/6    16/8      20/10    24/8       30/10    32/16
                 12/4/2  16/4/2    20/10/5  24/12      30/15    32/8/4
                 12/6/3  16/8/4             24/6/3     30/6/3   32/16/4
                         16/8/4/2           24/8/4     30/10/5  32/16/8
                                            24/12/4    30/15/5  32/8/4/2
                                            24/12/6             32/16/4/2
                                            24/8/4/2            32/16/8/4
                                            24/12/4/2           32/16/8/4/2
                                            24/12/6/3
The a(n) ordered factorizations for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:
  2  4    8      12     16       20     24       30     32
     2*2  4*2    4*3    4*4      5*4    6*4      6*5    8*4
          2*2*2  6*2    8*2      10*2   8*3      10*3   16*2
                 2*2*3  2*2*4    5*2*2  12*2     15*2   4*2*4
                 3*2*2  4*2*2           3*2*4    3*2*5  4*4*2
                        2*2*2*2         4*2*3    5*2*3  8*2*2
                                        4*3*2    5*3*2  2*2*2*4
                                        6*2*2           2*2*4*2
                                        2*2*2*3         4*2*2*2
                                        2*2*3*2         2*2*2*2*2
                                        3*2*2*2

Alois P. Heinz, Table of n, a(n) for n = 1..65536

The restriction to powers of 2 is A045690.
The inferior version is A337135.
The strictly inferior version is A342083.
The strictly superior version is A342084.
The additive version is A342094, with strict case A342095.
The additive version not allowing equality is A342098.
A001055 counts factorizations.
A003238 counts divisibility chains summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908, A341676.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A064052/A048098, A140271, A238535, A341673.
Cf. A000203, A001248, A005117, A006530, A020639, A057567, A057568, A112798, A169594, A337105, A342096, A342097.

a:= proc(n) option remember; 1+add(`if`(d>=n/d,
      a(d), 0), d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..128);  # Alois P. Heinz, Jun 24 2021

cmo[n_]:=Prepend[Prepend[#,n]&/@Join@@cmo/@Select[Most[Divisors[n]],#>=n/#&],{n}];
Table[Length[cmo[n]],{n,100}]

a(2^n) = A045690(n).

A350837 Number of integer partitions of n with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 14, 18, 24, 31, 41, 53, 70, 87, 112, 140, 178, 221, 277, 344, 428, 526, 648, 792, 971, 1180, 1436, 1738, 2103, 2533, 3049, 3660, 4387, 5242, 6259, 7450, 8860, 10511, 12453, 14723, 17387, 20489, 24121, 28343, 33269, 38982, 45632, 53327

Offset: 0

Gus Wiseman, Jan 18 2022

nonn

The first of these partitions that is not double-free (see A323092 for definition) is (4,3,2).

			The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (32)     (33)      (43)
                    (31)    (41)     (51)      (52)
                    (1111)  (311)    (222)     (61)
                            (11111)  (411)     (322)
                                     (3111)    (331)
                                     (111111)  (511)
                                               (4111)
                                               (31111)
                                               (1111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version with quotients >= 2 is A000929, sets A018819.
                           <= 2 is A342094, ranked by A342191.
                           < 2 is A342096, sets A045690, strict A342097.
                           > 2 is A342098, sets A040039.
The sets version (subsets of prescribed maximum) is A045691.
These partitions are ranked by A350838.
The strict case is A350840.
A version for differences is A350842, strict A350844.
The complement is counted by A350846, ranked by A350845.
A000041 = integer partitions.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free partitions, ranked by A320340.
Cf. A000070, A003000, A003114, `A003242, A051424, `A101417, A120641, A154402, A305148, A323093, A323094, A342095, A350839.

Table[Length[Select[IntegerPartitions[n], FreeQ[Divide@@@Partition[#,2,1],2]&]],{n,0,15}]

A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Jan 18 2022

nonn

Differs from A320340 in having 105: (4,3,2), 315: (4,3,2,2), 455: (6,4,3), etc.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with no adjacent prime indices of quotient 1/2.

			The terms and their prime indices begin:
      1: {}            19: {8}             38: {1,8}
      2: {1}           20: {1,1,3}         39: {2,6}
      3: {2}           22: {1,5}           40: {1,1,1,3}
      4: {1,1}         23: {9}             41: {13}
      5: {3}           25: {3,3}           43: {14}
      7: {4}           26: {1,6}           44: {1,1,5}
      8: {1,1,1}       27: {2,2,2}         45: {2,2,3}
      9: {2,2}         28: {1,1,4}         46: {1,9}
     10: {1,3}         29: {10}            47: {15}
     11: {5}           31: {11}            49: {4,4}
     13: {6}           32: {1,1,1,1,1}     50: {1,3,3}
     14: {1,4}         33: {2,5}           51: {2,7}
     15: {2,3}         34: {1,7}           52: {1,1,6}
     16: {1,1,1,1}     35: {3,4}           53: {16}
     17: {7}           37: {12}            55: {3,5}

The version with quotients >= 2 is counted by A000929, sets A018819.
                           <= 2 is A342191, counted by A342094.
                           < 2 is counted by A342096, sets A045690.
                           > 2 is counted by A342098, sets A040039.
The sets version (subsets of prescribed maximum) is counted by A045691.
These partitions are counted by A350837.
The strict case is counted by A350840.
For differences instead of quotients we have A350842, strict A350844.
The complement is A350845, counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000302, A001105, A003000, A018819, A094537, A120641, A154402, A319613, A323093, A337135, A342097, A342095.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[FreeQ[Divide@@@Partition[primeptn[#],2,1],2],{i,2,PrimeOmega[#]}]&]

A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 7, 8, 10, 13, 17, 19, 22, 25, 30, 35, 43, 52, 60, 70, 81, 93, 106, 122, 142, 166, 190, 216, 249, 287, 325, 371, 420, 479, 543, 617, 695, 784, 888, 1000, 1126, 1266, 1420, 1594, 1792, 2008, 2247, 2514, 2809, 3135, 3496, 3891, 4332

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(1) = 1 through a(13) = 13 partitions (A..D = 10..13):
  1   2   3   4    5    6    7    8     9     A     B     C     D
              31   32   51   43   53    54    64    65    75    76
                   41        52   62    72    73    74    93    85
                             61   71    81    82    83    A2    94
                                  431   432   91    92    B1    A3
                                        531   532   A1    543   B2
                                              541   641   651   C1
                                                    731   732   643
                                                          741   652
                                                          831   751
                                                                832
                                                                931
                                                                5431

The version for subsets of prescribed maximum is A045691.
The double-free case is A120641.
The non-strict case is A350837, ranked by A350838.
An additive version (differences) is A350844, non-strict A350842.
The non-strict complement is counted by A350846, ranked by A350845.
Versions for prescribed quotients:
  = 2:  A154402, sets A001511.
  != 2: A350840 (this sequence), sets A045691.
  >= 2: A000929, sets A018819.
  <= 2: A342095, non-strict A342094.
  < 2:  A342097, non-strict A342096, sets A045690.
  > 2:  A342098, sets A040039.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A003000, A018819, A303362, A323093, A323094, A337135, A342191.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[#[[i-1]]/#[[i]]!=2,{i,2,Length[#]}]&]],{n,0,30}]

A005434 Number of distinct autocorrelations of binary words of length n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 27, 30, 37, 47, 57, 62, 75, 87, 102, 116, 135, 155, 180, 194, 220, 254, 289, 312, 359, 392, 438, 479, 538, 595, 664, 701, 772, 863, 956, 1005, 1115, 1205, 1317, 1414, 1552, 1677, 1836, 1920, 2074, 2249, 2444

Offset: 1

Simon Plouffe, N. J. A. Sloane

Conjecture: a(n + 1) - a(n) < a(n + 13) - a(n + 12) for all n >= 1. - Eric Rowland, Nov 24 2021
From Eric Rivals, Jul 11 2023: (Start)
log(a(n))/log^2(n) converges when n tends to infinity. This conjecture was first stated in (Guibas and Odlyzko, JCTA, 1981a). (Rivals et al. ICALP 2023) proves this conjecture and provides an improved upper bound for this ratio.
An autocorrelation is a binary encoding of the period set.
This sequence is also the number of autocorrelation for words over any finite alphabet whose cardinality is at least two. The autocorrelation is independent of the alphabet cardinality, provided the cardinality is at least two; see proofs in (Guibas and Odlyzko, JCTA, 1981a). (End)

			From _Eric Rowland_, Nov 22 2021: (Start)
For n = 5 there are a(5) = 6 distinct autocorrelations of length-5 binary words:
  00000 can overlap itself in 1, 2, 3, 4, or 5 letters. Its autocorrelation is 11111.
  00100 can overlap itself in 1, 2, or 5 letters. Its autocorrelation is 10011.
  01010 can overlap itself in 1, 3, or 5 letters. Its autocorrelation is 10101.
  00010 can overlap itself in 1 or 5 letters. Its autocorrelation is 10001.
  01001 can overlap itself in 2 or 5 letters. Its autocorrelation is 10010.
  00001 can only overlap itself in 5 letters. Its autocorrelation is 10000.
(End)

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publ., 2nd Ed., 1994. Section 8.4: Flipping Coins
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

E-Hern Lee, Table of n, a(n) for n = 1..654
L. J. Guibas, Periodicities in Strings, Combinatorial Algorithms on Words 1985, NATO ASI Vol. F12, 257-269
L. J. Guibas and A. M. Odlyzko, Periods in Strings, Journal of Combinatorial Theory A 30:1 (1981) 19-42.
Leo J. Guibas and Andrew M. Odlyzko, String overlaps, pattern matching and nontransitive games, Journal of Combinatorial Theory Series A, 30 (March 1981), 183-208.
H. Harborth, Endliche 0-1-Folgen mit gleichen Teilblöcken, Journal für Mathematik, 271 (1974) 139-154.
A. Kaseorg, Rust program used to compute values for n up to 500
Eric Rivals, Incremental computation of the set of period sets, arXiv:2410.12077 [cs.DM], 2024. Refers to this sequence.
E. Rivals, and S. Rahmann (2001). Combinatorics of Periods in Strings. In: Orejas, F., Spirakis, P.G., van Leeuwen, J.(eds) Automata, Languages and Programming. ICALP 2001. Lecture Notes in Computer Science, vol 2076. Springer, Berlin, Heidelberg. doi:10.1007/3-540-48224-5_51.
E. H. Rivals and S. Rahmann, Combinatorics of Periods in Strings, Journal of Combinatorial Theory - Series A, Vol. 104(1) (2003), pp. 95-113.
E. H. Rivals, Autocorrelation of Strings.
E. H. Rivals and S. Rahmann Combinatorics of Periods in Strings
E. Rivals, M. Sweering, and P. Wang, Convergence of the Number of Period Sets in Strings. 50th International Colloquium on Automata, Languages, and Programming, {ICALP} 2023, Leibniz International Proceedings in Informatics (LIPIcs), ICALP 2023: 100:1-100:14. doi:10.4230/LIPIcs.ICALP.2023.100
T. Sillke, Autocorrelation Range
T. Sillke, kappa sequence for words of length n
T. Sillke, The autocorrelation function

Cf. A018819 (related to a lower bound for autocorrelations), A045690 (the number of binary strings sharing the same autocorrelation).

A005434 := proc( n :: posint )
    local    S := table();
    for local c in Iterator:-BinaryGrayCode( n ) do
        c := convert( c, 'list' );
        S[ [seq]( evalb( c[ 1 .. i + 1 ] = c[ n - i .. n ] ), i = 0 .. n - 1 ) ] := 0
    end do;
    numelems( S )
end proc: # James McCarron, Jun 21 2017

Table[Length[Union[Map[Flatten[Position[Table[Take[#,n-i]==Drop[#,i],{i,0,n-1}],True]-1]&,Tuples[{0,1},n]]]],{n,1,15}] (* Geoffrey Critzer, Nov 29 2013 *)

More terms and additional references from torsten.sillke(at)lhsystems.com

Definition clarified by Eric Rowland, Nov 22 2021

A007148 Number of self-complementary 2-colored bracelets (turnover necklaces) with 2n beads.

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 37, 74, 143, 284, 559, 1114, 2206, 4394, 8740, 17418, 34696, 69194, 137971, 275280, 549258, 1096286, 2188333, 4369162, 8724154, 17422652, 34797199, 69505908, 138845926, 277383872, 554189329, 1107297290, 2212558942

Offset: 1

N. J. A. Sloane

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations Pacific J. Math., 110 (1984), 203-221.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to bracelets

Cf. A000013, A000079, A007147.

Different from, but easily confused with, A045690 and A093371.

# see A245558
L := proc(n,k)
    local a,j ;
    a := 0 ;
    for j in numtheory[divisors](igcd(n,k)) do
        a := a+numtheory[mobius](j)*binomial(n/j,k/j) ;
    end do:
    a/n ;
end proc:
A007148 := proc(n)
    local a,k,l;
    a := 0 ;
    for k from 1 to n do
        for l in numtheory[divisors](igcd(n,k)) do
            a := a+L(n/l,k/l)*ceil(k/2/l) ;
        end do:
    end do:
    a;
end proc:
seq(A007148(n),n=1..20) ; # R. J. Mathar, Jul 23 2017

a[n_] := (1/2)*(2^(n-1) + Total[ EulerPhi[2*#]*2^(n/#) &  /@ Divisors[n]]/(2*n)); Table[ a[n], {n, 1, 33}] (* Jean-François Alcover, Oct 25 2011 *)

a(n)= (1/2) *(2^(n-1)+sumdiv(n,k,eulerphi(2*k)*2^(n/k))/(2*n))

from sympy import divisors, totient
def a(n):
    if n==1: return 1
    return 2**(n - 2) + sum(totient(2*d)*2**(n//d) for d in divisors(n))//(4*n)
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Jul 24 2017

a(n) = 2^(n-2) + (1/(4n)) * Sum_{d|n} phi(2d)*2^(n/d). - N. J. A. Sloane, Sep 25 2012

a(n) = (1/2)*(A000079(n-1) + A000013(n)).

Description corrected by Christian G. Bower

A045691 Number of binary words of length n with autocorrelation function 2^(n-1)+1.

Original entry on oeis.org

0, 1, 1, 3, 5, 11, 19, 41, 77, 159, 307, 625, 1231, 2481, 4921, 9883, 19689, 39455, 78751, 157661, 315015, 630337, 1260049, 2520723, 5040215, 10081661, 20160841, 40324163, 80643405, 161291731, 322573579, 645157041, 1290294393, 2580608475, 5161177495

Offset: 0

Torsten Sillke (torsten.sillke(AT)lhsystems.com)

nonn

From Gus Wiseman, Jan 22 2022: (Start)
Also the number of subsets of {1..n} containing n but without adjacent elements of quotient 1/2. The Heinz numbers of these sets are a subset of the squarefree terms of A320340. For example, the a(1) = 1 through a(6) = 19 subsets are:
  {1}  {2}  {3}    {4}      {5}        {6}
            {1,3}  {1,4}    {1,5}      {1,6}
            {2,3}  {3,4}    {2,5}      {2,6}
                   {1,3,4}  {3,5}      {4,6}
                   {2,3,4}  {4,5}      {5,6}
                            {1,3,5}    {1,4,6}
                            {1,4,5}    {1,5,6}
                            {2,3,5}    {2,5,6}
                            {3,4,5}    {3,4,6}
                            {1,3,4,5}  {3,5,6}
                            {2,3,4,5}  {4,5,6}
                                       {1,3,4,6}
                                       {1,3,5,6}
                                       {1,4,5,6}
                                       {2,3,4,6}
                                       {2,3,5,6}
                                       {3,4,5,6}
                                       {1,3,4,5,6}
                                       {2,3,4,5,6}
(End)

If a(n) counts subsets of {1..n} with n and without adjacent quotients 1/2:
- The version with quotients <= 1/2 is A018819, partitions A000929.
- The version with quotients < 1/2 is A040039, partitions A342098.
- The version with quotients >= 1/2 is A045690(n+1), partitions A342094.
- The version with quotients > 1/2 is A045690, partitions A342096.
- Partitions of this type are counted by A350837, ranked by A350838.
- Strict partitions of this type are counted by A350840.
- For differences instead of quotients we have A350842, strict A350844.
- Partitions not of this type are counted by A350846, ranked by A350845.
A000740 = relatively prime subsets of {1..n} containing n.
A002843 = compositions with all adjacent quotients >= 1/2.
A050291 = double-free subsets of {1..n}.
A154402 = partitions with all adjacent quotients 2.
A308546 = double-closed subsets of {1..n}, with maximum: shifted right.
A323092 = double-free integer partitions, ranked by A320340, strict A120641.
A326115 = maximal double-free subsets of {1..n}.
Cf. A000009, A001511, A003000, A003114, A116932, A274199, A323093, A342095, A342191, A342331, A342332, A342333, A342337.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&And@@Table[#[[i-1]]/#[[i]]!=1/2,{i,2,Length[#]}]&]],{n,0,15}] (* Gus Wiseman, Jan 22 2022 *)

a(2*n-1) = 2*a(2*n-2) - a(n) for n >= 2; a(2*n) = 2*a(2*n-1) + a(n) for n >= 2.

More terms from Sean A. Irvine, Mar 18 2021

A342339 Heinz numbers of the integer partitions counted by A342337, which have all adjacent parts (x, y) satisfying either x = y or x = 2y.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 36, 37, 41, 42, 43, 47, 48, 49, 53, 54, 59, 61, 63, 64, 65, 67, 71, 72, 73, 79, 81, 83, 84, 89, 96, 97, 101, 103, 107, 108, 109, 113, 121, 125, 126, 127, 128, 131, 133, 137

Offset: 1

Gus Wiseman, Mar 11 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}            19: {8}             48: {1,1,1,1,2}
      2: {1}           21: {2,4}           49: {4,4}
      3: {2}           23: {9}             53: {16}
      4: {1,1}         24: {1,1,1,2}       54: {1,2,2,2}
      5: {3}           25: {3,3}           59: {17}
      6: {1,2}         27: {2,2,2}         61: {18}
      7: {4}           29: {10}            63: {2,2,4}
      8: {1,1,1}       31: {11}            64: {1,1,1,1,1,1}
      9: {2,2}         32: {1,1,1,1,1}     65: {3,6}
     11: {5}           36: {1,1,2,2}       67: {19}
     12: {1,1,2}       37: {12}            71: {20}
     13: {6}           41: {13}            72: {1,1,1,2,2}
     16: {1,1,1,1}     42: {1,2,4}         73: {21}
     17: {7}           43: {14}            79: {22}
     18: {1,2,2}       47: {15}            81: {2,2,2,2}

The first condition alone gives A000961 (perfect powers).
The second condition alone is counted by A154402.
These partitions are counted by A342337.
A018819 counts partitions into powers of 2.
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A045690 counts sets with maximum n in with adjacent elements y < 2x.
A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
A342330 counts compositions with x < 2y and y < 2x (strict: A342341).
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
A342334 counts compositions with adjacent parts x >= 2y or y > 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
A342342 counts strict compositions with adjacent parts x <= 2y and y <= 2x.
Cf. A003114, A003242, A034296, A040039, A167606. A342083, A342084, A342087, A342191, A342336, A342339, A342340.

Select[Range[100],With[{y=PrimePi/@First/@FactorInteger[#]},And@@Table[y[[i]]==y[[i-1]]||y[[i]]==2*y[[i-1]],{i,2,Length[y]}]]&]

A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 144, 147, 150, 156, 162, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258, 260, 264, 266, 270

Offset: 1

Gus Wiseman, Jan 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at least two adjacent prime indices of quotient 1/2.

			The terms and corresponding partitions begin:
   6: (2,1)
  12: (2,1,1)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  30: (3,2,1)
  36: (2,2,1,1)
  42: (4,2,1)
  48: (2,1,1,1,1)
  54: (2,2,2,1)
  60: (3,2,1,1)
  63: (4,2,2)
  65: (6,3)
  66: (5,2,1)
  72: (2,2,1,1,1)
  78: (6,2,1)
  84: (4,2,1,1)
  90: (3,2,2,1)
  96: (2,1,1,1,1,1)

The complement is A350838, counted by A350837.
The strict complement is counted by A350840.
These partitions are counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A325160 ranks strict partitions with no successions, counted by A003114.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],MemberQ[Divide@@@Partition[primeptn[#],2,1],2]&]

A350846 Number of integer partitions of n with at least two adjacent parts of quotient 2.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 8, 12, 18, 25, 36, 48, 65, 89, 119, 157, 207, 269, 350, 448, 574, 729, 927, 1166, 1465, 1830, 2282, 2827, 3501, 4309, 5300, 6483, 7923, 9641, 11718, 14187, 17155, 20674, 24885, 29860, 35787, 42772, 51054, 60791, 72289, 85772, 101641

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(3) = 1 through a(9) = 12 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (521)      (621)
                       (2211)   (3211)    (3221)     (3321)
                       (21111)  (22111)   (4211)     (4221)
                                (211111)  (22211)    (5211)
                                          (32111)    (22221)
                                          (221111)   (32211)
                                          (2111111)  (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)

The complement is counted by A350837, strict A350840.
The complimentary additive version is A350842, strict A350844.
These partitions are ranked by A350845, complement A350838.
A000041 = integer partitions.
A323092 = double-free integer partitions, ranked by A320340.
Cf. A000929, A003000, A003114, A018819, A045690, A045691, A116931, A120641, A154402, A323093, A342094, A342095, A342096, A342098.

Table[Length[Select[IntegerPartitions[n], MemberQ[Divide@@@Partition[#,2,1],2]&]],{n,0,30}]

cp's OEIS Frontend

A342085 Number of decreasing chains of distinct superior divisors starting with n.

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A350837 Number of integer partitions of n with no adjacent parts of quotient 2.

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A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

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A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

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A005434 Number of distinct autocorrelations of binary words of length n.

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A007148 Number of self-complementary 2-colored bracelets (turnover necklaces) with 2n beads.

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A045691 Number of binary words of length n with autocorrelation function 2^(n-1)+1.

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A342339 Heinz numbers of the integer partitions counted by A342337, which have all adjacent parts (x, y) satisfying either x = y or x = 2y.

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