cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Gus Wiseman, May 18 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A325755.

Examples

			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)

Crossrefs

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

Programs

  • Mathematica
    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}]

A352486 Heinz numbers of non-self-conjugate integer partitions.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73
Offset: 1

Views

Author

Gus Wiseman, Mar 20 2022

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions. The sequence lists all Heinz numbers of partitions whose Heinz number is different from that of their conjugate.

Examples

			The terms together with their prime indices begin:
   3: (2)
   4: (1,1)
   5: (3)
   7: (4)
   8: (1,1,1)
  10: (3,1)
  11: (5)
  12: (2,1,1)
  13: (6)
  14: (4,1)
  15: (3,2)
  16: (1,1,1,1)
  17: (7)
  18: (2,2,1)
For example, the self-conjugate partition (4,3,3,1) has Heinz number 350, so 350 is not in the sequence.

Crossrefs

The complement is A088902, counted by A000700.
These partitions are counted by A330644.
These are the positions of nonzero terms in A352491.
A000041 counts integer partitions, strict A000009.
A098825 counts permutations by unfixed points.
A238349 counts compositions by fixed points, rank statistic A352512.
A325039 counts partitions w/ same product as conjugate, ranked by A325040.
A352523 counts compositions by unfixed points, rank statistic A352513.
Heinz number (rank) and partition:
- A003963 = product of partition, conjugate A329382
- A008480 = number of permutations of partition, conjugate A321648.
- A056239 = sum of partition
- A122111 = rank of conjugate partition
- A296150 = parts of partition, reverse A112798, conjugate A321649
- A352487 = less than conjugate, counted by A000701
- A352488 = greater than or equal to conjugate, counted by A046682
- A352489 = less than or equal to conjugate, counted by A046682
- A352490 = greater than conjugate, counted by A000701
Cf. A000720, A026424, A120383, A175508, A195017, A238745, A301987, A304360, A316524, A324846, A350841.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y0]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#!=Times@@Prime/@conj[primeMS[#]]&]

Formula

a(n) != A122111(a(n)).

A324696 Lexicographically earliest sequence containing 1 and all numbers divisible by prime(m) for some m not already in the sequence.

Original entry on oeis.org

1, 3, 6, 7, 9, 11, 12, 14, 15, 18, 19, 21, 22, 24, 27, 28, 29, 30, 33, 35, 36, 38, 39, 41, 42, 44, 45, 48, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 63, 66, 69, 70, 71, 72, 75, 76, 77, 78, 81, 82, 83, 84, 87, 88, 90, 91, 93, 95, 96, 97, 98, 99, 101, 102, 105
Offset: 1

Views

Author

Gus Wiseman, Mar 10 2019

Keywords

Comments

A self-describing sequence, similar to A304360.
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.

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   6: {1,2}
   7: {4}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}

Crossrefs

Complement of A304360.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098.
Cf. A324694, A324695, A324697, A324698, A324699, A324700, A324701, A324702, A324703, A324704, A324705.

Programs

  • Mathematica
    aQ[n_]:=n==1||Or@@Cases[FactorInteger[n],{p_,k_}:>!aQ[PrimePi[p]]];
Select[Range[100],aQ]

A324751 Number of strict integer partitions of n containing no prime indices of the parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 3, 2, 4, 5, 5, 6, 8, 8, 12, 10, 14, 13, 18, 19, 26, 25, 30, 34, 39, 40, 51, 55, 60, 71, 77, 90, 97, 111, 123, 136, 153, 170, 179, 216, 230, 264, 282, 322, 345, 385, 423, 470, 513, 573, 629, 686, 755, 834, 910, 1005, 1095, 1194, 1303, 1433
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2019

Keywords

Comments

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.

Examples

			The a(1) = 1 through a(13) = 8 strict integer partitions (A...D = 10...13):
  1   2   3   4    5   6    7    8    9    A    B     C     D
              31       42   43   71   54   64   65    75    76
                       51   52        63   73   83    84    85
                                      72   82   542   93    94
                                           91   731   A2    B2
                                                      B1    643
                                                            751
                                                            931

Crossrefs

The subset version is A324741, with maximal case A324743. The non-strict version is A324756. The Heinz number version is A324758. An infinite version is A304360.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713, A306844, A324764.
Cf. A324695, A324742, A324748.

Programs

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

Views

Author

Gus Wiseman, Mar 13 2019

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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.

Programs

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

Extensions

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

A324572 Number of integer partitions of n whose multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in decreasing order.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 3, 0, 2, 0, 4, 1, 2, 1, 4, 1, 3, 1, 5, 3, 5, 1, 6, 2, 6, 1, 7, 2, 7, 2, 11, 4, 8, 3, 11, 5, 10, 4, 13, 5, 11, 5, 16, 8, 14, 5, 19, 8, 18, 6, 22, 8, 22, 7, 26, 10, 25, 8, 33, 12, 29, 11, 36, 13, 34, 12, 40, 16, 41, 14, 47, 17, 45, 16, 55
Offset: 0

Views

Author

Gus Wiseman, Mar 08 2019

Keywords

Comments

These are a kind of self-describing partitions (cf. A001462, A304679).
The Heinz numbers of these partitions are given by A324571.
The case where the distinct parts are taken in increasing order is counted by A033461, with Heinz numbers given by A109298.

Examples

			The first 19 terms count the following integer partitions:
   1: (1)
   4: (22)
   4: (211)
   6: (3111)
   8: (41111)
   9: (333)
  10: (511111)
  10: (322111)
  12: (6111111)
  12: (4221111)
  12: (33222)
  14: (71111111)
  14: (52211111)
  16: (811111111)
  16: (622111111)
  16: (4444)
  16: (442222)
  17: (43331111)
  18: (9111111111)
  18: (7221111111)
  19: (533311111)

Crossrefs

Cf. A001156, A033461, A109298, A117144, A276078, A324524, A324571.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Union[#]==Length/@Split[#]&]],{n,0,30}]

Extensions

More terms from Alois P. Heinz, Mar 08 2019

A324704 Lexicographically earliest sequence containing 1 and all numbers > 2 divisible by prime(m) for some m already in the sequence.

Original entry on oeis.org

1, 4, 6, 7, 8, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 84
Offset: 1

Views

Author

Gus Wiseman, Mar 11 2019

Keywords

Comments

A self-describing sequence, similar to A304360.
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.

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   4: {1,1}
   6: {1,2}
   7: {4}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}

Crossrefs

Complement of A324698.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324699, A324700, A324701, A324702, A324703, A324705.

Programs

A324764 Number of anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 9, 20, 41, 89, 196, 443, 987, 2246, 5114, 11757, 27122, 62898, 146392, 342204, 802429, 1887882
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets are (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}}}}
{{{o}},{o,{o}}}

Examples

			The a(1) = 1 through a(7) = 9 anti-transitive rooted identity trees:
  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))))))

Links

Crossrefs

Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.
Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.

Programs

  • Mathematica
    idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],Intersection[Union@@#,#]=={}&]],{n,10}]

Extensions

a(21)-a(22) from Jinyuan Wang, Jun 20 2020

A324849 Positive integers divisible by none of their prime indices > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			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}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}

Links

Crossrefs

Complement of A324771.
Cf. A003963, A056239, A112798, A120383, A289509, A304360, A306844.
Cf. A324695, A324741, A324743, A324756, A324758, A324765, A324846, A324847, A324848, A324850, A324852, A324853, A324856.

Programs

  • Maple
    filter:= proc(n) andmap(t -> not ((n/numtheory:-pi(t))::integer), numtheory:-factorset(n) minus {2}) end proc:
select(filter, [$1..200]); # Robert Israel, Mar 20 2019
  • Mathematica
    Select[Range[100],!Or@@Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>If[p==2,False,Divisible[#,PrimePi[p]]]]&]
  • PARI
    is(n) = my(f=factor(n)[, 1]~, idc=[]); for(k=1, #f, idc=concat(idc, [primepi(f[k])])); for(t=1, #idc, if(idc[t]==1, next); if(n%idc[t]==0, return(0))); 1 \\ Felix Fröhlich, Mar 21 2019

A324698 Lexicographically earliest sequence containing 2 and all numbers > 1 whose prime indices already belong to the sequence.

Original entry on oeis.org

2, 3, 5, 9, 11, 15, 23, 25, 27, 31, 33, 45, 47, 55, 69, 75, 81, 83, 93, 97, 99, 103, 115, 121, 125, 127, 135, 137, 141, 155, 165, 197, 207, 211, 225, 235, 243, 249, 253, 257, 275, 279, 291, 297, 309, 341, 345, 347, 363, 375, 379, 381, 405, 411, 415, 419, 423
Offset: 1

Views

Author

Gus Wiseman, Mar 10 2019

Keywords

Comments

A self-describing sequence, similar to A304360.
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.

Examples

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   5: {3}
   9: {2,2}
  11: {5}
  15: {2,3}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  31: {11}
  33: {2,5}
  45: {2,2,3}
  47: {15}
  55: {3,5}
  69: {2,9}
  75: {2,3,3}
  81: {2,2,2,2}
  83: {23}
  93: {2,11}
  97: {25}
  99: {2,2,5}

Crossrefs

Complement of A324704.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324699, A324700, A324701, A324702, A324703, A324705.

Programs

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