A000837 -id:A000837 - cp's OEIS frontend

A305148 Number of integer partitions of n whose distinct parts are pairwise indivisible.

1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 12, 12, 17, 20, 22, 28, 35, 39, 48, 55, 65, 79, 90, 105, 121, 143, 166, 190, 219, 254, 290, 332, 382, 436, 493, 567, 637, 729, 824, 931, 1052, 1186, 1334, 1504, 1691, 1894, 2123, 2380, 2664, 2968, 3319, 3704, 4119, 4586, 5110

Offset: 0

Gus Wiseman, May 26 2018

nonn

			The a(9) = 7 integer partitions are (9), (72), (54), (522), (333), (3222), (111111111).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..360 (terms 0..300 from Alois P. Heinz)

Cf. A000837, A001055, A001970, A007359, A007716, A051424, A078374, A100953, A285572, A285573, A302696, A303362, A305079, A305149, A305150.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],2],UnsameQ@@#&&Divisible@@#&]=={}&]],{n,20}]

More terms from Alois P. Heinz, May 26 2018

A317081 Number of integer partitions of n whose multiplicities cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 5, 9, 11, 16, 20, 30, 34, 50, 58, 79, 96, 129, 152, 203, 243, 307, 375, 474, 563, 707, 850, 1042, 1246, 1532, 1815, 2215, 2632, 3173, 3765, 4525, 5323, 6375, 7519, 8916, 10478, 12414, 14523, 17133, 20034, 23488, 27422, 32090, 37285, 43511, 50559

Offset: 0

Gus Wiseman, Jul 21 2018

nonn

Also the number of integer partitions of n with distinct section-sums, where the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. - Gus Wiseman, Apr 21 2025

			The a(1) = 1 through a(9) = 16 partitions:
 (1) (2) (3)  (4)   (5)   (6)   (7)    (8)    (9)
         (21) (31)  (32)  (42)  (43)   (53)   (54)
              (211) (41)  (51)  (52)   (62)   (63)
                    (221) (321) (61)   (71)   (72)
                    (311) (411) (322)  (332)  (81)
                                (331)  (422)  (432)
                                (421)  (431)  (441)
                                (511)  (521)  (522)
                                (3211) (611)  (531)
                                       (3221) (621)
                                       (4211) (711)
                                              (3321)
                                              (4221)
                                              (4311)
                                              (5211)
                                              (32211)

Chai Wah Wu, Table of n, a(n) for n = 0..160

The case with parts also covering an initial interval is A317088.
These partitions are ranked by A317090.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047966 counts partitions with constant section-sums.
A048767 interchanges prime indices and prime multiplicities (Look-and-Say), see A048768.
A055932 lists numbers whose prime indices cover an initial interval.
A116540 counts normal set multipartitions.
A304442 counts partitions with equal run-sums, ranks A353833.
A381436 lists the section-sum partition of prime indices.
A381440 lists the Look-and-Say partition of prime indices.
Cf. A000837, A069799, A217605, A317082, A317084, A317085, A317087.
Cf. A003242, A116861, A239455, A353837, A364916, A381431, A381432, A381438.

normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],normalQ[Length/@Split[#]]&]],{n,30}]

from sympy.utilities.iterables import partitions
def A317081(n):
    if n == 0:
        return 1
    c = 0
    for d in partitions(n):
        s = set(d.values())
        if len(s) == max(s):
            c += 1
    return c # Chai Wah Wu, Jun 22 2020

A328171 Number of (necessarily strict) integer partitions of n with no two consecutive parts divisible.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 4, 5, 4, 9, 9, 10, 12, 14, 16, 20, 23, 29, 34, 38, 41, 51, 60, 66, 78, 89, 103, 119, 137, 157, 180, 201, 229, 261, 298, 338, 379, 431, 486, 547, 618, 694, 783, 876, 986, 1103, 1241, 1387, 1551, 1728, 1932, 2148, 2395, 2664, 2963

Offset: 0

Gus Wiseman, Oct 11 2019

nonn

			The a(1) = 1 through a(15) = 10 partitions (A..F = 10..15):
  1  2  3  4  5   6  7   8   9    A    B   C    D    E     F
              32     43  53  54   64   65  75   76   86    87
                     52      72   73   74  543  85   95    96
                             432  532  83  732  94   A4    B4
                                       92       A3   B3    D2
                                                B2   653   654
                                                643  743   753
                                                652  752   852
                                                832  5432  A32
                                                           6432

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..330

The complement is counted by A328221.
The Heinz numbers of these partitions are A328603.
Partitions whose pairs of consecutive parts are relatively prime are A328172, with strict case A328188.
Partitions with no pair of consecutive parts relatively prime are A328187, with strict case A328220.
Numbers without consecutive divisible proper divisors are A328028.
Cf. A000837, A018783, A328026, A328161, A328189, A328194, A328195.

Table[Length[Select[IntegerPartitions[n],!MatchQ[#,{_,x_,y_,_}/;Divisible[x,y]]&]],{n,0,30}]

A007562 Number of planted trees where non-root, non-leaf nodes an even distance from root are of degree 2.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 36, 72, 137, 275, 541, 1098, 2208, 4521, 9240, 19084, 39451, 82113, 171240, 358794, 753460, 1587740, 3353192, 7100909, 15067924, 32044456, 68272854, 145730675, 311575140, 667221030, 1430892924, 3072925944, 6607832422, 14226665499

Offset: 1

N. J. A. Sloane

There is no planted tree on one node by definition.
Column k=2 of A144018. - Alois P. Heinz, Oct 17 2012
It appears that a(n) is also the number of locally non-intersecting unlabeled rooted trees with n nodes, where a tree is locally non-intersecting if the branches directly under of any non-leaf node have empty intersection. - Gus Wiseman, Aug 22 2018

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 10*x^7 + 20*x^8 + 36*x^9 + ...
From _Joerg Arndt_, Jun 23 2014: (Start)
The a(8) = 20 such trees have the following level sequences:
01:  [ 0 1 2 3 4 3 2 1 ]
02:  [ 0 1 2 3 3 3 2 1 ]
03:  [ 0 1 2 3 3 2 2 1 ]
04:  [ 0 1 2 3 3 2 1 1 ]
05:  [ 0 1 2 3 2 3 2 1 ]
06:  [ 0 1 2 3 2 2 2 1 ]
07:  [ 0 1 2 3 2 2 1 1 ]
08:  [ 0 1 2 3 2 1 2 1 ]
09:  [ 0 1 2 3 2 1 1 1 ]
10:  [ 0 1 2 2 2 2 2 1 ]
11:  [ 0 1 2 2 2 2 1 1 ]
12:  [ 0 1 2 2 2 1 2 1 ]
13:  [ 0 1 2 2 2 1 1 1 ]
14:  [ 0 1 2 2 1 2 2 1 ]
15:  [ 0 1 2 2 1 2 1 1 ]
16:  [ 0 1 2 2 1 1 1 1 ]
17:  [ 0 1 2 1 2 1 2 1 ]
18:  [ 0 1 2 1 2 1 1 1 ]
19:  [ 0 1 2 1 1 1 1 1 ]
20:  [ 0 1 1 1 1 1 1 1 ]
Successive levels change by at most 1 and the last level is 1, compare to the example in A000081.
(End)
From _Gus Wiseman_, Aug 22 2018: (Start)
The a(7) = 10 locally non-intersecting trees:
  (o(o(oo)))
  (o(oo(o)))
  (o(oooo))
  (oo(o(o)))
  (oo(ooo))
  (o(o)(oo))
  (ooo(oo))
  (oo(o)(o))
  (oooo(o))
  (oooooo)
(End)

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

Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A000081, A000837, A004111, A007560, A144018, A289509, A316470, A316473, A316475.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d= divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): a:= n-> `if`(n<=1, n, b(n-2)): seq(a(n), n=1..40);  # Alois P. Heinz, Sep 06 2008

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, (Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}] + Sum[ d*p[d], {d, Divisors[n]}])/n]; b]; b = etr[a]; a[n_] := If[n <= 1, n, b[n-2]]; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)
purt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Intersection@@#=={}&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,10}] (* Gus Wiseman, Aug 22 2018 *)

{a(n) = local(A); if( n<2, n>0, A = x / (1 - x) + O(x^n); for(k=2, n-2, A /= (1 - x^k + O(x^n))^polcoeff(A, k-1)); polcoeff(A, n-1))}; /* Michael Somos, Oct 06 2003 */

Shifts left 2 places under Euler transform.
G.f.: x + x^2 / (Product_{k>0} (1 - x^k)^a(k)). - Michael Somos, Oct 06 2003
a(n) ~ c * d^n / n^(3/2), where d = 2.246066877341161662499621547921... and  c = 0.68490297576105466417608032... . - Vaclav Kotesovec, Jun 23 2014
G.f. A(x) satisfies: A(x) = x + x^2 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...). - Ilya Gutkovskiy, Jun 11 2021

Better description from Christian G. Bower, May 15 1998

A304711 Heinz numbers of integer partitions whose distinct parts are pairwise coprime.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 80, 82, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106, 108, 110

Offset: 1

Gus Wiseman, May 17 2018

nonn

Two parts are coprime if they have no common divisor greater than 1. For partitions of length 1 note that (1) is coprime but (x) is not coprime for x > 1.

First differs from A289509 at a(24) = 44, A289509(24) = 42.

			Sequence of all partitions whose distinct parts are pairwise coprime begins (1), (11), (21), (111), (31), (211), (41), (32), (1111), (221), (311), (51), (2111), (61), (411), (321), (11111), (52), (71), (43), (2211), (81), (3111).

Cf. A000837, A007359, A018783, A051424, A056239, A078374, A101268, A289508, A289509, A298748, A300486, A302569, A302696, A302698, A302796, A302797, A304709.

Select[Range[200],CoprimeQ@@PrimePi/@FactorInteger[#][[All,1]]&]

A303707 Number of factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

First differs from A081707 at a(60) = 9, A081707(60) = 8.

			The a(60) = 9 factorizations are (2*2*3*5), (2*2*15), (2*3*10), (2*5*6), (2*30), (3*20), (5*12), (6*10), (60).

Cf. A000837, A001055, A001597, A007716, A007916, A045778, A052409, A052410, A162247, A281113, A281116, A303708, A303709, A303710.

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

Dirichlet g.f.: Product_{n in A007916} 1/(1 - n^s).

A324756 Number of integer partitions of n containing no prime indices of the parts.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 7, 7, 9, 11, 16, 16, 24, 25, 34, 39, 50, 54, 70, 79, 96, 111, 135, 152, 186, 208, 249, 285, 335, 377, 448, 506, 588, 664, 777, 873, 1010, 1139, 1309, 1471, 1697, 1890, 2175, 2435, 2772, 3106, 3532, 3941, 4478, 4995, 5643, 6297, 7107, 7897

Offset: 0

Gus Wiseman, Mar 17 2019

nonn

These could be described as anti-transitive integer partitions.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(1) = 1 through a(8) = 9 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (43)       (44)
                    (31)    (11111)  (42)      (52)       (71)
                    (1111)           (51)      (331)      (422)
                                     (222)     (511)      (2222)
                                     (3111)    (31111)    (3311)
                                     (111111)  (1111111)  (5111)
                                                          (311111)
                                                          (11111111)

The subset version is A324741, with maximal case A324743. The strict case is A324751. The Heinz number version is A324758. An infinite version is A324695.

Cf. A000720, A000837, A001462, A051424, A112798, A276625, A304360, A306844, A324764, A324742, A324753.

Table[Length[Select[IntegerPartitions[n],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]

A328673 Number of integer partitions of n in which no two distinct parts are relatively prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 2, 15, 2, 17, 10, 23, 2, 39, 2, 46, 18, 58, 2, 95, 8, 103, 31, 139, 2, 219, 3, 232, 59, 299, 22, 452, 4, 492, 104, 645, 5, 920, 5, 1006, 204, 1258, 8, 1785, 21, 1994, 302, 2442, 11, 3366, 71, 3738, 497, 4570, 18, 6253, 24, 6849

Offset: 0

Gus Wiseman, Oct 29 2019

nonn

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

			The a(1) = 1 through a(10) = 9 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        63         55
              1111         42               62        333        64
                           222              422       111111111  82
                           111111           2222                 442
                                            11111111             622
                                                                 4222
                                                                 22222
                                                                 1111111111

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

The Heinz numbers of these partitions are A328867 (strict case is A318719).
The relatively prime case is A328672.
The strict case is A318717.
The version for non-isomorphic multiset partitions is A319752.
The version for set-systems is A305843.
The version involving all parts (not just distinct ones) is A200976.
Cf. A000837, A202425, A305148, A305854, A306006, A316476, A326910.

Table[Length[Select[IntegerPartitions[n],And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]&]],{n,0,20}]

a(n > 0) = A200976(n) + 1.

A200976 Number of partitions of n such that each pair of parts (if any) has a common factor.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 14, 1, 16, 9, 22, 1, 38, 1, 45, 17, 57, 1, 94, 7, 102, 30, 138, 1, 218, 2, 231, 58, 298, 21, 451, 3, 491, 103, 644, 4, 919, 4, 1005, 203, 1257, 7, 1784, 20, 1993, 301, 2441, 10, 3365, 70, 3737, 496, 4569, 17, 6252, 23, 6848

Offset: 0

Alois P. Heinz, Nov 29 2011

nonn

a(n) is different from A018783(n) for n = 0, 31, 37, 41, 43, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, ... .

Every pair of (possibly equal) parts has a common factor > 1. These partitions are said to be (pairwise) intersecting. - Gus Wiseman, Nov 04 2019

			a(0) = 1: [];
a(4) = 2: [2,2], [4];
a(9) = 3: [3,3,3], [3,6], [9];
a(31) = 2: [6,10,15], [31];
a(41) = 4: [6,10,10,15], [6,15,20], [6,14,21], [41].

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350 (terms 0..250 from Alois P. Heinz)
L. Naughton, G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8

Cf. A018783.
The version with only distinct parts compared is A328673.
The relatively prime case is A202425.
The strict case is A318717.
The version for non-isomorphic multiset partitions is A319752.
The version for set-systems is A305843.
Cf. A000837, A305148, A305854, A306006, A316476, A328672, A328867, A328868.

b:= proc(n, j, s) local ok, i;
      if n=0 then 1
    elif j<2 then 0
    else ok:= true;
         for i in s while ok do ok:= evalb(igcd(i, j)<>1) od;
         `if`(ok, add(b(n-j*k, j-1, [s[], j]), k=1..n/j), 0) +b(n, j-1, s)
      fi
    end:
a:= n-> b(n, n, []):
seq(a(n), n=0..62);

b[n_, j_, s_] := Module[{ok, i, is}, Which[n == 0, 1, j < 2, 0, True, ok = True; For[is = 1, is <= Length[s] && ok, is++, i = s[[is]]; ok = GCD[i, j] != 1]; If[ok, Sum[b[n-j*k, j-1, Append[s, j]], {k, 1, n/j}], 0] + b[n, j-1, s]]]; a[n_] := b[n, n, {}]; Table[a[n], {n, 0, 62}] (* Jean-François Alcover, Dec 26 2013, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],And[And@@(GCD[##]>1&)@@@Select[Tuples[Union[#],2],LessEqual@@#&]]&]],{n,0,20}] (* Gus Wiseman, Nov 04 2019 *)

a(n > 0) = A328673(n) - 1. - Gus Wiseman, Nov 04 2019

A303546 Number of non-isomorphic aperiodic multiset partitions of weight n.

Original entry on oeis.org

1, 3, 9, 29, 90, 285, 909, 2984, 9935, 34113, 119368, 428923, 1574223, 5915235, 22699730, 89000042, 356058539, 1453069854, 6044132793, 25612564200, 110503626702, 485161228675, 2166488899641, 9835209480533, 45370059225227

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is aperiodic if its multiplicities are relatively prime. For this sequence neither the parts nor their multiset union are required to be aperiodic, only the multiset of parts.

			Non-isomorphic representatives of the a(3) = 9 aperiodic multiset partitions are:
  {{1,1,1}}, {{1,2,2}}, {{1,2,3}},
  {{1},{1,1}}, {{1},{2,2}}, {{1},{2,3}}, {{2},{1,2}},
  {{1},{2},{2}}, {{1},{2},{3}}.

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A000740, A000837, A001055, A007716, A007916, A049311, A100953, A255906, A269134, A275024, A293993, A301700, A303386, A303431, A303547.

a(n) = Sum_{d|n} mu(d) * A007716(n/d).

cp's OEIS Frontend

A305148 Number of integer partitions of n whose distinct parts are pairwise indivisible.

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A317081 Number of integer partitions of n whose multiplicities cover an initial interval of positive integers.

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A328171 Number of (necessarily strict) integer partitions of n with no two consecutive parts divisible.

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A007562 Number of planted trees where non-root, non-leaf nodes an even distance from root are of degree 2.

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A304711 Heinz numbers of integer partitions whose distinct parts are pairwise coprime.

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A303707 Number of factorizations of n using elements of A007916 (numbers that are not perfect powers).

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A324756 Number of integer partitions of n containing no prime indices of the parts.

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A328673 Number of integer partitions of n in which no two distinct parts are relatively prime.

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