A324753 -id:A324753 - cp's OEIS frontend

A237984 Number of partitions of n whose mean is a part.

1, 2, 2, 3, 2, 5, 2, 6, 5, 8, 2, 21, 2, 14, 22, 30, 2, 61, 2, 86, 67, 45, 2, 283, 66, 80, 197, 340, 2, 766, 2, 663, 543, 234, 703, 2532, 2, 388, 1395, 4029, 2, 4688, 2, 4476, 7032, 1005, 2, 17883, 2434, 9713, 7684, 14472, 2, 25348, 17562, 37829, 16786, 3721

Offset: 1

Clark Kimberling, Feb 27 2014

a(n) = 2 if and only if n is a prime.

			a(6) counts these partitions:  6, 33, 321, 222, 111111.
From _Gus Wiseman_, Sep 14 2019: (Start)
The a(1) = 1 through a(10) = 8 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      432        22222
                           321              3221      531        32221
                           111111           4211      111111111  33211
                                            11111111             42211
                                                                 52111
                                                                 1111111111
(End)

Cf. A238478.
The Heinz numbers of these partitions are A327473.
A similar sequence for subsets is A065795.
Dominated by A067538.
The strict case is A240850.
Partitions without their mean are A327472.
Cf. A000016, A316413, A324753, A325705, A327478, A327482.

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Mean[p]]], {n, 40}]

from sympy.utilities.iterables import partitions
def A237984(n): return sum(1 for s,p in partitions(n,size=True) if not n%s and n//s in p) # Chai Wah Wu, Sep 21 2023

a(n) = A000041(n) - A327472(n). - Gus Wiseman, Sep 14 2019

A290689 Number of transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 88, 143, 229, 370, 592, 955, 1527, 2457, 3929, 6304, 10081

Offset: 1

Gus Wiseman, Oct 19 2017

A rooted tree is transitive if every proper terminal subtree is also a branch of the root. First differs from A206139 at a(13) = 143.

Regarding the notation, a rooted tree is a finite multiset of rooted trees. For example, the rooted tree (o(o)(oo)) is short for {{},{{}},{{},{}}}. Each "o" is a leaf. Each pair of parentheses corresponds to a non-leaf node (such as the root). Its contents "(...)" represent a branch. - Gus Wiseman, Nov 16 2024

			The a(7) = 8 7-node transitive rooted trees are: (o(oooo)), (oo(ooo)), (o(o)((o))), (o(o)(oo)), (ooo(oo)), (oo(o)(o)), (oooo(o)), (oooooo).

Robert P. P. McKone, The transitive rooted trees with n=1 to n=22 nodes.

The restriction to identity trees (A004111) is A279861, ranks A290760.
These trees are ranked by A290822.
The anti-transitive version is A306844, ranks A324758.
The totally transitive case is A318185 (by leaves A318187), ranks A318186.
A version for integer partitions is A324753, for subsets A324736.
The ordered version is A358453, ranks A358457, undirected A358454.
Cf. A000081, A001192, A001678, A324770, A324969, A325705, A358455, A358460.

nn=18;
rtall[n_]:=If[n===1,{{}},Module[{cas},Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])]]];
Table[Length[Select[rtall[n],Complement[Union@@#,#]==={}&]],{n,nn}]

a(20) from Robert Price, Sep 13 2018

a(21)-a(22) from Robert P. P. McKone, Dec 16 2023

A324758 Heinz numbers of integer partitions containing no prime indices of the parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 37, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 57, 59, 61, 62, 63, 64, 65, 67, 68, 71, 73, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101

Offset: 1

Gus Wiseman, Mar 17 2019

nonn

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 Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
These could be described as anti-transitive numbers (cf. A290822), as they are numbers x such that if prime(y) divides x and prime(z) divides y, then prime(z) does not divide x.
Also numbers n such that A003963(n) is coprime to n.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  27: {2,2,2}

The subset version is A324741, with maximal case A324743. The strict integer partition version is A324751. The integer partition version is A324756. An infinite version is A324695.

Cf. A000720, A001221, A001462, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324742, A324753, A324764.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[primeMS[#],Union@@primeMS/@primeMS[#]]=={}&]

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}]

A325702 Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 2, 1, 3, 3, 8, 7, 10, 13, 17, 19, 28, 35, 38, 51, 67, 81, 100, 128, 157, 195, 233, 285, 348, 427, 506, 613, 733, 873, 1063, 1263, 1503, 1802, 2131, 2537, 3005, 3565, 4171, 4922, 5820, 6775, 8001, 9333, 10860, 12739, 14840, 17206, 20029, 23248

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325755.

			The partition x = (4,3,1,1,1) has multiplicities (3,1,1), which are a submultiset of x, so x is counted under a(10).
The a(1) = 1 through a(11) = 7 partitions:
  (1)  (22)   (221)  (2211)  (3211)  (4211)   (333)    (3322)    (7211)
       (211)         (3111)          (32111)  (5211)   (3331)    (33221)
                                     (41111)  (32211)  (6211)    (52211)
                                                       (42211)   (53111)
                                                       (43111)   (322211)
                                                       (322111)  (332111)
                                                       (421111)  (431111)
                                                       (511111)

Cf. A000041, A181819, A225486, A290689, A290822, A304360, A323014, A324736, A324748, A324753, A324843, A325254, A325755.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap]
Table[Length[Select[IntegerPartitions[n],submultQ[Sort[Length/@Split[#]],#]&]],{n,0,30}]

A324736 Number of subsets of {1...n} containing all prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 15, 22, 43, 79, 127, 175, 343, 511, 851, 1571, 3141, 4397, 8765, 13147, 25243, 46843, 76795, 115171, 230299, 454939, 758203, 1516363, 2916079, 4356079, 8676079, 12132079, 24264157, 45000157, 73800253, 145685053, 291369853, 437054653, 728424421

Offset: 0

Gus Wiseman, Mar 13 2019

nonn

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.

Also the number of subsets of {1...n} containing no prime indices of the non-elements up to n.

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,4}      {1,4}        {1,4}
                           {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,6}
                                      {1,2,3,5}    {1,2,3,4}
                                      {1,2,3,4,5}  {1,2,3,5}
                                                   {1,2,3,6}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,3,4,5,6}
An example for n = 18 is {1,2,4,7,8,9,12,16,17,18}, whose elements have the following prime indices:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
All of these prime indices {1,2,4,7} belong to the subset, as required.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The strict integer partition version is A324748. The integer partition version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A304360, A320426.
Cf. A324697, A324737, A324741, A324743.

Table[Length[Select[Subsets[Range[n]],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,10}]

pset(n)={my(b=0, f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 15 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 15 2019

A325705 Number of integer partitions of n containing all of their distinct multiplicities.

Original entry on oeis.org

1, 1, 0, 1, 3, 2, 4, 3, 7, 8, 16, 15, 24, 28, 39, 44, 68, 80, 98, 130, 167, 200, 259, 320, 396, 497, 601, 737, 910, 1107, 1335, 1631, 1983, 2372, 2887, 3439, 4166, 4949, 5940, 7043, 8450, 9980, 11884, 13984, 16679, 19493, 23162, 27050, 31937, 37334, 43926

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325706.

			The partition (4,2,1,1,1,1) has distinct multiplicities {1,4}, both of which belong to the partition, so it is counted under a(10).
The a(0) = 1 through a(10) = 16 partitions:
  ()  (1)  (21)  (22)   (41)   (51)    (61)    (71)     (81)     (91)
                 (31)   (221)  (321)   (421)   (431)    (333)    (541)
                 (211)         (2211)  (3211)  (521)    (531)    (631)
                               (3111)          (3221)   (621)    (721)
                                               (4211)   (3321)   (3322)
                                               (32111)  (4221)   (3331)
                                               (41111)  (5211)   (4321)
                                                        (32211)  (5221)
                                                                 (6211)
                                                                 (32221)
                                                                 (33211)
                                                                 (42211)
                                                                 (43111)
                                                                 (322111)
                                                                 (421111)
                                                                 (511111)

Cf. A109297, A114639, A114640, A181819, A225486, A290689, A324753, A324843, A325702, A325706, A325707, A325755.

Table[Length[Select[IntegerPartitions[n],SubsetQ[Sort[#],Sort[Length/@Split[#]]]&]],{n,0,30}]

A324843 Number of unlabeled rooted trees with n nodes where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 9, 15, 17, 31, 35, 57, 70, 111, 136, 213, 265, 405, 517, 763, 987, 1458, 1893, 2736, 3611, 5161, 6836, 9702

Offset: 1

Gus Wiseman, Mar 18 2019

A subset of totally transitive rooted trees (A318185).

			The a(1) = 1 through a(8) = 8 rooted trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)    (ooooooo)
                (o(o))  (oo(o))  (oo(oo))   (ooo(oo))   (ooo(ooo))
                                 (ooo(o))   (oooo(o))   (oooo(oo))
                                 (o(o)(o))  (oo(o)(o))  (ooooo(o))
                                                        (oo(o)(oo))
                                                        (ooo(o)(o))
                                                        (o(o)(o)(o))
                                                        (o(o)(o(o)))

The Matula-Goebel numbers of these trees are given by A324842.
Cf. A000081, A279861, A290689, A290822, A318185.
Cf. A324704, A324736, A324748, A324753, A324847, A324848, A324854.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])],And@@Table[submultQ[b,#],{b,#}]&];
Table[Length[rallt[n]],{n,10}]

A324748 Number of strict integer partitions of n containing all prime indices of the parts.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 3, 2, 2, 4, 3, 4, 3, 5, 6, 9, 8, 7, 8, 11, 12, 13, 15, 17, 22, 22, 20, 28, 31, 32, 36, 41, 43, 53, 53, 59, 70, 76, 77, 89, 99, 108, 124, 135, 139, 160, 172, 188, 209, 229, 243, 274, 298, 315, 353, 391, 417, 457, 496, 538, 588

Offset: 0

Gus Wiseman, Mar 15 2019

nonn

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 first 15 terms count the following integer partitions.
   1: (1)
   3: (2,1)
   5: (4,1)
   6: (3,2,1)
   7: (4,2,1)
   9: (8,1)
   9: (6,2,1)
  10: (4,3,2,1)
  11: (8,2,1)
  11: (5,3,2,1)
  12: (9,2,1)
  12: (7,4,1)
  12: (6,3,2,1)
  13: (8,4,1)
  13: (6,4,2,1)
  14: (8,3,2,1)
  14: (7,4,2,1)
  15: (12,2,1)
  15: (9,3,2,1)
  15: (8,4,2,1)
  15: (5,4,3,2,1)
An example for n = 6 is (20,18,11,5,3,2,1), with prime indices:
  20: {1,1,3}
  18: {1,2,2}
  11: {5}
   5: {3}
   3: {2}
   2: {1}
   1: {}
All of these prime indices {1,2,3,5} belong to the partition, as required.

The subset version is A324736. The non-strict version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A279861, A290689, A290760, A305713.
Cf. A324697, A324737, A324751.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,30}]

A324737 Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 6, 8, 16, 24, 48, 84, 168, 216, 432, 648, 1296, 2448, 4896, 6528, 13056, 19584, 39168, 77760, 155520, 229248, 458496, 790272, 1580544, 3128832, 6257664, 9386496, 18772992, 24081408, 48162816, 95938560, 191877120, 378335232, 756670464, 1135005696, 2270011392

Offset: 1

Gus Wiseman, Mar 13 2019

nonn

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.

Also the number of subsets of {2...n} with complement containing no term whose prime indices all belong to the subset.

			The a(1) = 1 through a(6) = 16 subsets:
  {}  {}   {}     {}       {}         {}
      {2}  {3}    {3}      {4}        {4}
           {2,3}  {4}      {5}        {5}
                  {2,3}    {3,5}      {6}
                  {3,4}    {4,5}      {3,5}
                  {2,3,4}  {2,3,5}    {4,5}
                           {3,4,5}    {4,6}
                           {2,3,4,5}  {5,6}
                                      {2,3,5}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}
                                      {2,3,4,5}
                                      {2,3,5,6}
                                      {3,4,5,6}
                                      {2,3,4,5,6}
An example for n = 15 is {2, 3, 5, 8, 9, 10, 11, 15}. The numbers from 2 to 15 with all prime indices in the subset are {3, 5, 9, 11, 15}, which all belong to the subset, as required.

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A290689, A290822, A304360, A306844.

Cf. A324697, A324698, A324736, A324738, A324748, A324753, A324755.

Table[Length[Select[Subsets[Range[2,n]],Function[set,SubsetQ[set,Select[Range[2,n],SubsetQ[set,PrimePi/@First/@FactorInteger[#]]&]]]]],{n,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1, k, pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k, b)->if(k>#p, 1, my(t=self()(k+1, b+(1<Andrew Howroyd, Aug 24 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 24 2019

cp's OEIS Frontend

A237984 Number of partitions of n whose mean is a part.

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A290689 Number of transitive rooted trees with n nodes.

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A324758 Heinz numbers of integer partitions containing no prime indices of the parts.

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

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A325702 Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).

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A324736 Number of subsets of {1...n} containing all prime indices of the elements.

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A325705 Number of integer partitions of n containing all of their distinct multiplicities.

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A324843 Number of unlabeled rooted trees with n nodes where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

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