A167865 -id:A167865 - cp's OEIS frontend

A342087 Number of chains of divisors starting with n and having no adjacent parts x <= y^2.

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 8, 2, 4, 4, 8, 2, 10, 2, 6, 6, 4, 2, 12, 2, 6, 4, 6, 2, 10, 4, 8, 4, 4, 2, 14, 2, 4, 6, 6, 4, 10, 2, 6, 4, 8, 2, 16, 2, 4, 6, 6, 4, 10, 2, 12, 4, 4, 2, 14

Offset: 1

Gus Wiseman, Mar 05 2021

nonn

An alternative wording: Number of chains of divisors starting with n and having all adjacent parts x > y^2.

			The chains for n = 1, 2, 6, 12, 24, 42, 48:
   1    2      6        12        24        42          48
        2/1    6/1      12/1      24/1      42/1        48/1
               6/2      12/2      24/2      42/2        48/2
               6/2/1    12/3      24/3      42/3        48/3
                        12/2/1    24/4      42/6        48/4
                        12/3/1    24/2/1    42/2/1      48/6
                                  24/3/1    42/3/1      48/2/1
                                  24/4/1    42/6/1      48/3/1
                                            42/6/2      48/4/1
                                            42/6/2/1    48/6/1
                                                        48/6/2
                                                        48/6/2/1

Amiram Eldar, Table of n, a(n) for n = 1..10000

The restriction to powers of 2 is A018819.
Not requiring strict inferiority gives A067824.
The weakly inferior version is twice A337135.
The case ending with 1 is counted by A342083.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A342098, or A000929 allowing equality.
A000005 counts divisors, with sum A000203.
A001055 counts factorizations.
A003238 counts chains of divisors 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 ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A334997 counts chains of divisors of n by length.
Cf. A002033, A006530, A018819, A020639, A337105, A341674, A342086, A342094, A342095, A342097.

cem[n_]:=Prepend[Prepend[#,n]&/@Join@@cem/@Most[Divisors[n]],{n}];
Table[Length[Select[cem[n],And@@Thread[Divide@@@Partition[#,2,1]>Rest[#]]&]],{n,30}]

For n > 1, a(n) = 2*A342083(n).

A083711 a(n) = A083710(n) - A000041(n-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 5, 3, 7, 1, 14, 1, 13, 8, 20, 1, 33, 1, 40, 14, 44, 1, 85, 6, 79, 25, 117, 1, 181, 1, 196, 45, 233, 17, 389, 1, 387, 80, 545, 1, 750, 1, 839, 165, 1004, 1, 1516, 12, 1612, 234, 2040, 1, 2766, 48, 3142, 388, 3720, 1, 5295, 1, 5606, 663, 7038, 83, 9194, 1, 10379, 1005

Offset: 1

N. J. A. Sloane, Jun 16 2003

Number of integer partitions of n with no 1's with a part dividing all the others. If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 18 2021

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(6) = 4 through a(12) = 13 partitions:
  (6)      (7)  (8)        (9)      (10)         (11)  (12)
  (3,3)         (4,4)      (6,3)    (5,5)              (6,6)
  (4,2)         (6,2)      (3,3,3)  (8,2)              (8,4)
  (2,2,2)       (4,2,2)             (4,4,2)            (9,3)
                (2,2,2,2)           (6,2,2)            (10,2)
                                    (4,2,2,2)          (4,4,4)
                                    (2,2,2,2,2)        (6,3,3)
                                                       (6,4,2)
                                                       (8,2,2)
                                                       (3,3,3,3)
                                                       (4,4,2,2)
                                                       (6,2,2,2)
                                                       (4,2,2,2,2)
                                                       (2,2,2,2,2,2)
(End)

L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Allowing 1's gives A083710.
The strict case is A098965.
The complement (except also without 1's) is counted by A338470.
The dual version is A339619.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A001787, A001792, A015723, A097986, A098743, A130689, A130714, A264401, A339563, A342193.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-1 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 2 to 100 do printf(`%d,`,a(j)) od: # James Sellers, Jun 21 2003
# second Maple program:
a:= n-> max(1, add(combinat[numbpart](d-1), d=numtheory[divisors](n) minus {n})):
seq(a(n), n=1..69);  # Alois P. Heinz, Feb 15 2023

a[n_] := If[n==1, 1, Sum[PartitionsP[d-1], {d, Most@Divisors[n]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 15 2023 *)

a(n) = Sum_{ d|n, dA000041(d-1).

More terms from James Sellers, Jun 21 2003

A163767 a(n) = tau_{n}(n) = number of ordered n-factorizations of n.

Original entry on oeis.org

1, 2, 3, 10, 5, 36, 7, 120, 45, 100, 11, 936, 13, 196, 225, 3876, 17, 3078, 19, 4200, 441, 484, 23, 62400, 325, 676, 3654, 11368, 29, 27000, 31, 376992, 1089, 1156, 1225, 443556, 37, 1444, 1521, 459200, 41, 74088, 43, 43560, 46575, 2116, 47, 11995200, 1225

Offset: 1

Paul D. Hanna, Aug 04 2009

nonn

Also the number of length n - 1 chains of divisors of n. - Gus Wiseman, May 07 2021

			Successive Dirichlet self-convolutions of the all 1's sequence begin:
(1),1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,... (A000012)
1,(2),2,3,2,4,2,4,3,4,2,6,2,4,4,5,... (A000005)
1,3,(3),6,3,9,3,10,6,9,3,18,3,9,9,15,... (A007425)
1,4,4,(10),4,16,4,20,10,16,4,40,4,16,16,35,... (A007426)
1,5,5,15,(5),25,5,35,15,25,5,75,5,25,25,70,... (A061200)
1,6,6,21,6,(36),6,56,21,36,6,126,6,36,36,126,... (A034695)
1,7,7,28,7,49,(7),84,28,49,7,196,7,49,49,210,... (A111217)
1,8,8,36,8,64,8,(120),36,64,8,288,8,64,64,330,... (A111218)
1,9,9,45,9,81,9,165,(45),81,9,405,9,81,81,495,... (A111219)
1,10,10,55,10,100,10,220,55,(100),10,550,10,100,... (A111220)
1,11,11,66,11,121,11,286,66,121,(11),726,11,121,... (A111221)
1,12,12,78,12,144,12,364,78,144,12,(936),12,144,... (A111306)
...
where the main diagonal forms this sequence.
From _Gus Wiseman_, May 07 2021: (Start)
The a(1) = 1 through a(5) = 5 chains of divisors:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)
      (2)  (3/1)  (2/1/1)  (5/1/1/1)
           (3/3)  (2/2/1)  (5/5/1/1)
                  (2/2/2)  (5/5/5/1)
                  (4/1/1)  (5/5/5/5)
                  (4/2/1)
                  (4/2/2)
                  (4/4/1)
                  (4/4/2)
                  (4/4/4)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Enrique Pérez Herrero)

Main diagonal of A077592.
Diagonal n = k + 1 of the array A334997.
The version counting all multisets of divisors (not just chains) is A343935.
A000005 counts divisors.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts nonempty strict chains of divisors of n.
A251683/A334996 count strict nonempty length-k divisor chains from n to 1.
A337255 counts strict length-k chains of divisors starting with n.
A339564 counts factorizations with a selected factor.
A343662 counts strict length-k chains of divisors (row sums: A337256).
Cf. A002033, A007425, A008480, A018818, A062319, A066959, A186972, A327527, A337105, A337107, A343658.
Cf. A060690.

Table[Times@@(Binomial[#+n-1,n-1]&/@FactorInteger[n][[All,2]]),{n,1,50}] (* Enrique Pérez Herrero, Dec 25 2013 *)

{a(n,m=n)=if(n==1,1,if(m==1,1,sumdiv(n,d,a(d,1)*a(n/d,m-1))))}

from math import prod, comb
from sympy import factorint
def A163767(n): return prod(comb(n+e-1,e) for e in factorint(n).values()) # Chai Wah Wu, Jul 05 2024

a(p) = p for prime p.
a(n) = n^k when n is the product of k distinct primes (conjecture).
a(n) = n-th term of the n-th Dirichlet self-convolution of the all 1's sequence.
a(2^n) = A060690(n). - Alois P. Heinz, Jun 12 2024

A343377 Number of strict integer partitions of n with no part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 8, 9, 13, 18, 21, 26, 32, 38, 47, 57, 66, 80, 95, 110, 132, 157, 181, 211, 246, 282, 327, 379, 435, 500, 570, 648, 743, 849, 963, 1094, 1241, 1404, 1592, 1799, 2025, 2282, 2568, 2882, 3239, 3634, 4066, 4554, 5094, 5686, 6346

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are empty or have greatest part not divisible by all the others.

			The a(5) = 1 through a(12) = 9 partitions:
  (3,2)  (3,2,1)  (4,3)  (5,3)    (5,4)    (6,4)      (6,5)      (7,5)
                  (5,2)  (4,3,1)  (7,2)    (7,3)      (7,4)      (5,4,3)
                         (5,2,1)  (4,3,2)  (5,3,2)    (8,3)      (6,4,2)
                                  (5,3,1)  (5,4,1)    (9,2)      (6,5,1)
                                           (7,2,1)    (5,4,2)    (7,3,2)
                                           (4,3,2,1)  (6,4,1)    (7,4,1)
                                                      (7,3,1)    (8,3,1)
                                                      (5,3,2,1)  (9,2,1)
                                                                 (5,4,2,1)

The dual strict complement is A097986.
The dual version is A341450.
The non-strict version is A343341 (Heinz numbers: A343337).
The strict complement is counted by A343347.
The case with smallest part not divisible by all the others is A343379.
The case with smallest part divisible by all the others is A343381.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A130689, A200745, A264401, A338470, A339562, A343338, A343342, A343345, A343346, A343382.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 5, 3, 9, 9, 12, 12, 18, 18, 27, 27, 36, 41, 51, 51, 73, 80, 96, 105, 132, 137, 177, 188, 230, 253, 303, 320, 398, 431, 508, 550, 659, 705, 847, 913, 1063, 1165, 1359, 1452, 1716, 1856, 2134, 2329, 2688, 2894, 3345, 3622, 4133

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are either empty, or (1) have smallest part not dividing all the others and (2) have greatest part not divisible by all the others.

			The a(5) = 1 through a(13) = 9 partitions (empty column indicated by dot):
  (3,2)  .  (4,3)  (5,3)  (5,4)    (6,4)    (6,5)    (7,5)    (7,6)
            (5,2)         (7,2)    (7,3)    (7,4)    (5,4,3)  (8,5)
                          (4,3,2)  (5,3,2)  (8,3)    (7,3,2)  (9,4)
                                            (9,2)             (10,3)
                                            (5,4,2)           (11,2)
                                                              (6,4,3)
                                                              (6,5,2)
                                                              (7,4,2)
                                                              (8,3,2)

The first condition alone gives A341450.
The non-strict version is A343342  (Heinz numbers: A343338).
The second condition alone gives A343377.
The opposite version is A343378.
The half-opposite versions are A343380 and A343381.
The version for "or" instead of "and" is A343382.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A097986, A200745, A264401, A338470, A339562, A342193, A343337, A343341, A343343, A343346, A343347.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

The Heinz numbers for the non-strict version are A343338 = A342193 /\ A343337.

A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 3, 3, 5, 1, 7, 1, 8, 4, 6, 1, 15, 2, 9, 5, 14, 1, 22, 1, 20, 7, 18, 4, 36, 1, 26, 10, 40, 1, 51, 1, 48, 18, 49, 1, 86, 3, 73, 19, 86, 1, 117, 7, 120, 27, 120, 1, 196, 1, 160, 42, 201, 10, 259, 1, 258, 50, 292, 1, 407, 1, 357, 81, 431, 8, 548, 1, 577

Offset: 1

Vladeta Jovovic, Oct 23 2004

If n > 0, we can assume this part is the smallest. - Gus Wiseman, Apr 18 2021

Alois P. Heinz, Table of n, a(n) for n = 1..10000

The non-strict version with 1's allowed is A083710.
The non-strict version is A083711.
The version with 1's allowed is A097986.
The Heinz numbers of these partitions are the odd terms of A339563.
The non-strict dual is A339619.
The strict complement is counted by A341450.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A130689, A130714, A200745, A264401, A343345, A343347, A343378, A343381.

Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 92}], {k, 2, 92}]], x], {2, 81}] (* Robert G. Wilson v, Nov 01 2004 *)
Table[If[n==0,0,Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&]]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

a(n) = Sum_{d|n, dA025147(d-1).

G.f.: Sum_{k>=2} (x^k*Product_{i>=2}(1 + x^(k*i))).

More terms from Robert G. Wilson v, Nov 01 2004

Name shortened by Gus Wiseman, Apr 23 2021

A317875 Number of achiral free pure multifunctions with n unlabeled leaves.

Original entry on oeis.org

1, 1, 3, 9, 30, 102, 369, 1362, 5181, 20064, 79035, 315366, 1272789, 5185080, 21296196, 88083993, 366584253, 1533953100, 6449904138, 27238006971, 115475933202, 491293053093, 2096930378415, 8976370298886, 38528771056425, 165784567505325

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

An achiral free pure multifunction is either (case 1) the leaf symbol "o", or (case 2) a nonempty expression of the form h[g, ..., g], where h and g are both achiral free pure multifunctions.

			The first 4 terms count the following multifunctions.
o,
o[o],
o[o,o], o[o[o]], o[o][o],
o[o,o,o], o[o[o][o]], o[o[o[o]]], o[o[o,o]], o[o][o,o], o[o][o[o]], o[o][o][o], o[o,o][o], o[o[o]][o].

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317853.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.
Cf. A317882, A317883, A317884, A317885.

a[n_]:=If[n==1,1,Sum[a[n-k]*Sum[a[d],{d,Divisors[k]}],{k,n-1}]];
Array[a,12]

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*(sum(k=1, n-1, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-1, v[i]*sumdiv(n-i, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = Sum_{0 < k < n} a(n - k) * Sum_{d|k} a(d).
From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x + A(x) * Sum_{k>=1} A(x^k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x + (Sum_{n>=1} a(n)*x^n) * (Sum_{n>=1} a(n)*x^n/(1 - x^n)). (End)

A337256 Number of strict chains of divisors of n.

Original entry on oeis.org

2, 4, 4, 8, 4, 12, 4, 16, 8, 12, 4, 32, 4, 12, 12, 32, 4, 32, 4, 32, 12, 12, 4, 80, 8, 12, 16, 32, 4, 52, 4, 64, 12, 12, 12, 104, 4, 12, 12, 80, 4, 52, 4, 32, 32, 12, 4, 192, 8, 32, 12, 32, 4, 80, 12, 80, 12, 12, 4, 176, 4, 12, 32, 128, 12, 52, 4, 32, 12, 52

Offset: 1

Gus Wiseman, Aug 23 2020

nonn

			The a(n) chains for n = 1, 2, 4, 6, 8 (empty chains shown as 0):
  0  0    0      0      0
  1  1    1      1      1
     2    2      2      2
     2/1  4      3      4
          2/1    6      8
          4/1    2/1    2/1
          4/2    3/1    4/1
          4/2/1  6/1    4/2
                 6/2    8/1
                 6/3    8/2
                 6/2/1  8/4
                 6/3/1  4/2/1
                        8/2/1
                        8/4/1
                        8/4/2
                        8/4/2/1

A067824 is the case of chains starting with n (or ending with 1).
A074206 is the case of chains from n to 1.
A253249 is the nonempty case.
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A074206 counts chains of divisors from n to 1.
A122651 counts chains of divisors summing to n.
A167865 counts chains of divisors > 1 summing to n.
A334996 appears to count chains of divisors from n to 1 by length.
A337070 counts chains of divisors starting with A006939(n).
A337071 counts chains of divisors starting with n!.
A337255 counts chains of divisors starting with n by length.
Cf. A001221, A002033, A008480, A124010, A251683, A337105, A337107.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Divisors[n],!(Divisible[#1,#2]||Divisible[#2,#1])&]],{n,10}]

a(n) = A253249(n) + 1.

A343347 Number of strict integer partitions of n with a part divisible by all the others.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 6, 5, 4, 6, 6, 6, 8, 7, 7, 10, 9, 9, 12, 10, 8, 11, 11, 10, 14, 13, 11, 13, 12, 15, 20, 17, 15, 19, 19, 19, 22, 18, 17, 23, 22, 22, 28, 25, 24, 31, 28, 26, 32, 32, 30, 34, 32, 29, 37, 33, 27, 36, 33, 34, 44, 38, 36, 45, 45

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are empty or have greatest part divisible by all the others.

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
        21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
                    51  421  71  81   91   632  93    841  D1   C3
                                 621  631  821  A2    931  842  E1
                                                B1    A21       C21
                                                6321            8421

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The dual version is A097986 (non-strict: A083710).
The non-strict version is A130689 (Heinz numbers: complement of A343337).
The strict complement is counted by A343377.
The case with smallest part divisible by all the others is A343378.
The case with smallest part not divisible by all the others is A343380.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A064410, A098743, A200745, A264401, A339563, A341450, A343341, A343344, A343379, A343382.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m*prod(i=1, #u-1, 1 + x^u[i] + O(x^(n-m+1)))))} \\ Andrew Howroyd, Apr 17 2021

G.f.: 1 + Sum_{k>0} (x^k/(1 + x^k))*Product_{d|k} (1 + x^d). - Andrew Howroyd, Apr 17 2021

A320222 Number of unlabeled rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 39, 78, 161, 324, 658, 1316, 2657, 5314, 10668, 21347, 42777, 85554, 171290, 342580, 685498, 1371037, 2742733, 5485466, 10972351, 21944711, 43892080, 87784323, 175574004, 351148008, 702307038, 1404614076, 2809249582, 5618499824, 11237042426

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

This is a weaker condition than achirality (cf. A003238).

			The a(1) = 1 through a(6) = 18 rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (o(o))   (o(oo))    (o(ooo))
                 (((o)))  (oo(o))    (oo(oo))
                          (((oo)))   (ooo(o))
                          ((o)(o))   (((ooo)))
                          ((o(o)))   ((o(oo)))
                          (o((o)))   ((oo(o)))
                          ((((o))))  (o((oo)))
                                     (o(o)(o))
                                     (o(o(o)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     (((o(o))))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

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

Cf. A002541, A003238, A010766, A126656, A014668, A167865, A214577, A316782, A317099, A317100, A317712, A320230.

saue[n_]:=Sum[If[SameQ@@DeleteCases[ptn,1],If[DeleteCases[ptn,1]=={},1,saue[DeleteCases[ptn,1][[1]]]],0],{ptn,IntegerPartitions[n-1]}];
Table[saue[n],{n,15}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=2, n-1, (n-1)\k*v[k])); v} \\ Andrew Howroyd, Oct 26 2018

a(n) = 1 + Sum_{k = 2..n-1} floor((n-1)/k) * a(k).

a(n) ~ c * 2^n, where c = 0.3270422384018894564479397100499014525700668391191792769625407295138546463... - Vaclav Kotesovec, Sep 07 2019

cp's OEIS Frontend

A342087 Number of chains of divisors starting with n and having no adjacent parts x <= y^2.

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A083711 a(n) = A083710(n) - A000041(n-1).

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A163767 a(n) = tau_{n}(n) = number of ordered n-factorizations of n.

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A343377 Number of strict integer partitions of n with no part divisible by all the others.

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A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.

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A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.

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A317875 Number of achiral free pure multifunctions with n unlabeled leaves.

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