A355382 -id:A355382 - cp's OEIS frontend

A355386 Position of first appearance of n in A355382, where A355382(m) = number of divisors d of m such that bigomega(d) = omega(m); or a(n) = -1 if n does not appear in A355382.

1, 12, 36, 120, 180, 360, 840, 1260, 5400, 27000, 2520, 5040, 6300, 7560, 15120, 12600, 25200

Offset: 1

Gus Wiseman, Jul 02 2022

The first position of -1 appears to be 18, pointed out by Amiram Eldar.
The terms are not always increasing.
The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.

			The terms together with their prime indices begin:
      1: {}
     12: {1,1,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    180: {1,1,2,2,3}
    360: {1,1,1,2,2,3}
    840: {1,1,1,2,3,4}
   1260: {1,1,2,2,3,4}
   5400: {1,1,1,2,2,2,3,3}
  27000: {1,1,1,2,2,2,3,3,3}
   2520: {1,1,1,2,2,3,4}
   5040: {1,1,1,1,2,2,3,4}
   6300: {1,1,2,2,3,3,4}
   7560: {1,1,1,2,2,2,3,4}
  15120: {1,1,1,1,2,2,2,3,4}
The terms together with their divisors satisfying the condition begin:
      1:   1
     12:   4,   6
     36:   4,   6,   9
    120:   8,  12,  20,  30
    180:  12,  18,  20,  30,  45
    360:   8,  12,  18,  20,  30,  45
    840:  24,  40,  56,  60,  84, 140, 210
   1260:  36,  60,  84,  90, 126, 140, 210, 315
   5400:   8,  12,  18,  20,  27,  30,  45,  50,  75
  27000:   8,  12,  18,  20,  27,  30,  45,  50,  75, 125
   2520:  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   5040:  16,  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   6300:  36,  60,  84,  90, 100, 126, 140, 150, 210, 225, 315, 350, 525

These are the positions of first appearances in A355382, which is the version of A181591 without multiplicity.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 counts prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
A355383 counts cmpsbl. pairs of partitions with containment, comps. A355384.
Cf. A000712, A022811, A056239, A071625, A181819, A319910, A339006.

tf=Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,1000}];
Table[Position[tf,n][[1,1]],{n,Select[Union[tf],SubsetQ[tf,Range[#]]&]}]

A133494 Diagonal of the array of iterated differences of A047848.

Original entry on oeis.org

1, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883

Offset: 0

Paul Barry, Paul Curtz, Dec 23 2007

a(n) is the number of ways to choose a composition C, and then choose a composition of each part of C. - Geoffrey Critzer, Mar 19 2012
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the reptend length of 1/3^(n+1) in decimal. - Jianing Song, Nov 14 2018
Also the number of pairs of integer compositions, the first summing to n and the second with sum equal to the length of the first. If an integer composition is regarded as an arrow from sum to length, these are composable pairs, and the obvious composition operation founds a category of integer compositions. For example, we have (2,1,1,4) . (1,2,1) . (1,2) = (2,6), where dots represent the composition operation. The version without empty compositions is A000244. Composable triples are counted by 1 followed by A000302. The unordered version is A022811. - Gus Wiseman, Jul 14 2022

			From _Gus Wiseman_, Jul 15 2020: (Start)
The a(0) = 1 through a(3) = 9 ways to choose a composition of each part of a composition:
  ()  (1)  (2)      (3)
           (1,1)    (1,2)
           (1),(1)  (2,1)
                    (1,1,1)
                    (1),(2)
                    (2),(1)
                    (1),(1,1)
                    (1,1),(1)
                    (1),(1),(1)
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Index entries for linear recurrences with constant coefficients, signature (3).

The strict version is A336139.
Splittings of partitions are A323583.
Multiset partitions of partitions are A001970.
Partitions of each part of a partition are A063834.
Compositions of each part of a partition are A075900.
Strict partitions of each part of a strict partition are A279785.
Compositions of each part of a strict partition are A304961.
Strict compositions of each part of a composition are A307068.
Compositions of each part of a strict composition are A336127.
Cf. A000012, A000244, A000302, A001906, A006951, A011782, A022811, A071919.
Cf. A072706, A078008, A112467, A182105, A316245, A336128, A336135.
Cf. A339006, A355382, A355384, A355387.

[n eq 0 select 1 else 3^(n-1): n in [0..30]]; // G. C. Greubel, Nov 20 2023

a:= n-> ceil(3^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2020

CoefficientList[Series[(1 - 2 x)/(1 - 3 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 3^(Range[0, 30])] (* G. C. Greubel, Nov 20 2023 *)

a(n)=max(1,3^(n-1)) \\ Charles R Greathouse IV, Jul 07 2011

Vec((1-2*x)/(1-3*x) + O(x^100)) \\  Altug Alkan, Oct 30 2015

[(3^n + 2*int(n==0))//3 for n in range(31)] # G. C. Greubel, Nov 20 2023

Binomial transform of A078008. - Paul Curtz, Aug 04 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: (1 - 2*x)/(1 - 3*x).
a(n) = A000244(n-1), n > 0. (End)
From Philippe Deléham, Nov 13 2008: (Start)
a(n) = Sum_{k=0..n} A112467(n,k)*2^k.
a(n) = Sum_{k=0..n} A071919(n,k)*2^k. (End)
Let A(x) be the g.f. Then B(x) = x*A(x) satisfies B(x/(1-x)) = x/(1 - 2*B(x)). - Vladimir Kruchinin, Dec 05 2011
G.f.: 1/(1 - (Sum_{k>=1} (x/(1 - x))^k)). - Joerg Arndt, Sep 30 2012
For n > 0, a(n) = 2*(Sum_{k=0..n-1} a(k)) - 1 = 3^(n-1). - J. Conrad, Oct 29 2015
G.f.: 1 + x/(1 + x)*(1 + 4*x/(1 + 4*x)*(1 + 7*x/(1 + 7*x)*(1 + 10*x/(1 + 10*x)*(1 + .... - Peter Bala, May 27 2017
Invert transform of A011782(n) = 2^(n-1). Second invert transform of A000012. - Gus Wiseman, Jul 19 2020
a(n) = ceiling(3^(n-1)). - Alois P. Heinz, Jul 26 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: (2 + exp(3*x))/3.
a(n) = 3*a(n-1) for n > 1. (End)

Definition clarified by R. J. Mathar, Nov 11 2008

A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 16, 26, 42, 64, 100, 150, 224, 330, 482, 697, 999, 1418, 1996, 2794, 3879, 5355, 7343, 10018, 13583, 18338, 24618, 32917, 43790, 58043, 76591, 100716, 131906, 172194, 223966, 290423, 375318, 483668, 621368, 796138, 1017146

Offset: 0

Gus Wiseman, Jul 02 2022

nonn

If a partition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of partitions.

			The a(0) = 1 through a(5) = 10 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
                (11)(1)  (21)(21)  (31)(31)   (41)(41)
                         (111)(1)  (22)(2)    (32)(32)
                                   (211)(11)  (311)(11)
                                   (211)(21)  (311)(31)
                                   (1111)(1)  (221)(21)
                                              (221)(22)
                                              (2111)(11)
                                              (2111)(21)
                                              (11111)(1)

With multiplicity we have A339006.
The version for compositions is A355384.
The homogeneous version w/o containment is A355385, compositions A355388.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Splitting partitions: A072706, A316245, A317715, A318683, A318684, A319794, A323433, A323583, A336131-A336136.
Cf. A000009, A022811, A032020, A070933, A181591, A181819, A319910, A355382.

Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,IntegerPartitions[n]}],{n,0,15}]

A355388 Number of composable pairs (y, v) of integer compositions of n, where a composition is regarded as an arrow from the number of parts to the number of distinct parts.

Original entry on oeis.org

1, 1, 2, 6, 18, 58, 174, 536, 1656, 4947, 14800, 43157, 126572, 364070, 1039926, 2938898, 8223400, 22846370, 62930113, 172177400, 467002792, 1259736804, 3371190792, 8973530491, 23728305128, 62421018163, 163255839779, 424842462529, 1100006243934, 2834558927244, 7270915592897

Offset: 0

Gus Wiseman, Jul 02 2022

nonn

Being composable here means that the length of v equals the number of distinct parts in y.

			The a(0) = 1 through a(4) = 18 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)
                (11)(2)  (21)(21)  (31)(31)
                         (21)(12)  (31)(13)
                         (12)(21)  (31)(22)
                         (12)(12)  (13)(31)
                         (111)(3)  (13)(13)
                                   (13)(22)
                                   (22)(4)
                                   (211)(31)
                                   (211)(13)
                                   (211)(22)
                                   (121)(31)
                                   (121)(13)
                                   (121)(22)
                                   (112)(31)
                                   (112)(13)
                                   (112)(22)
                                   (1111)(4)

Alois P. Heinz, Table of n, a(n) for n = 0..800 (first 201 terms from Andrew Howroyd)

The case with containment is A032020.
The inhomogeneous version with containment is A355384, partitions A355383.
The version for partitions is A355385, with containment A000009.
A133494 counts compositions of each part of a composition, strict A336139.
A323583 counts splittings of partitions.
Cf. A001970, A022811, A063834, A070933, A316245, A319910, A355382.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      expand(add(b(n-i*j, i-1, p+j)/j!*`if`(j=0, 1, x), j=0..n/i))))
    end:
a:= n-> (p-> add(coeff(p, x, i)*binomial(n-1, i-1), i=0..degree(p)))(b(n$2, 0)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 01 2023

Table[Length[Select[Tuples[Join@@Permutations/@IntegerPartitions[n],2], Length[Union[#[[1]]]]==Length[#[[2]]]&]],{n,0,10}]

a(n) = {if(n==0, 1, my(s=0); forpart(p=n, p=Vec(p); my(S=Set(p)); s += binomial(n-1, #S-1)*(#p)!/prod(i=1, #S, my(c=#select(t->t==S[i], p)); c! )); s)} \\ Andrew Howroyd, Jan 01 2023

\\ for larger n.
a(n) = { local(Cache=Map());
  my(F(r,m,p,q) = my(key=[r,m,p,q], z); if(!mapisdefined(Cache, key, &z),
  z = if(m==0, if(r==0, p!*binomial(n-1, q-1)), self()(r, m-1, p, q) + sum(j=1, r\m, self()(r-j*m, min(m-1, r-j*m), p+j, q+1)/j!));
  mapput(Cache, key, z) ); z);
  if(n==0, 1, F(n, n, 0, 0))
} \\ Andrew Howroyd, Jan 01 2023

a(n) = Sum_{k>=1} binomial(n-1, k-1)*A235998(n, k) for n > 0. - Andrew Howroyd, Jan 01 2023

Terms a(14) and beyond from Andrew Howroyd, Jan 01 2023

A355387 Number of ways to choose a distinct subsequence of an integer composition of n.

Original entry on oeis.org

1, 2, 5, 14, 37, 98, 259, 682, 1791, 4697, 12303, 32196, 84199, 220087, 575067, 1502176, 3923117, 10244069, 26746171, 69825070, 182276806, 475804961, 1241965456, 3241732629, 8461261457, 22084402087, 57640875725, 150442742575, 392652788250, 1024810764496

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

By "distinct" we mean equal subsequences are counted only once. For example, the pair (1,1)(1) is counted only once even though (1) is a subsequence of (1,1) in two ways. The version with multiplicity is A025192.

			The a(3) = 14 pairings of a composition with a chosen subsequence:
  (3)()     (3)(3)
  (21)()    (21)(1)   (21)(2)    (21)(21)
  (12)()    (12)(1)   (12)(2)    (12)(12)
  (111)()   (111)(1)  (111)(11)  (111)(111)

Christian Sievers, Table of n, a(n) for n = 0..2000

For partitions we have A000712, composable A339006.
The homogeneous version is A011782, without containment A000302.
With multiplicity we have A025192, for partitions A070933.
The strict case is A032005.
The case of strict subsequences is A236002.
The composable case is A355384, homogeneous without containment A355388.
A075900 counts compositions of each part of a partition.
A304961 counts compositions of each part of a strict partition.
A307068 counts strict compositions of each part of a composition.
A336127 counts compositions of each part of a strict composition.
Cf. A011782, A022811, A032020, A063834, A133494, A181591, A323583, A331330, A336128, A336130, A336139, A355382, A355383.

Table[Sum[Length[Union[Subsets[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,6}]

lista(n)=my(f=sum(k=1,n,(x^k+x*O(x^n))/(1-x/(1-x)+x^k)));Vec((1-x)/((1-2*x)*(1-f))) \\ Christian Sievers, May 06 2025

G.f.: (1-x)/((1-2*x)*(1-f)) where f = Sum_{k>=1} x^k/(1-x/(1-x)+x^k) is the generating function for A331330. - Christian Sievers, May 06 2025

a(16) and beyond from Christian Sievers, May 06 2025

cp's OEIS Frontend

A355386 Position of first appearance of n in A355382, where A355382(m) = number of divisors d of m such that bigomega(d) = omega(m); or a(n) = -1 if n does not appear in A355382.

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A133494 Diagonal of the array of iterated differences of A047848.

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A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

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A355388 Number of composable pairs (y, v) of integer compositions of n, where a composition is regarded as an arrow from the number of parts to the number of distinct parts.

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A355387 Number of ways to choose a distinct subsequence of an integer composition of n.

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