A181821 -id:A181821 - cp's OEIS frontend

A353389 Create the sequence of all positive integers > 1 that are prime or whose prime shadow (A181819) is a divisor that is already in the sequence. Then remove all the primes.

9, 36, 125, 225, 441, 1089, 1260, 1521, 1980, 2340, 2401, 2601, 2772, 3060, 3249, 3276, 3420, 4140, 4284, 4761, 4788, 5148, 5220, 5580, 5796, 6660, 6732, 7308, 7380, 7524, 7569, 7740, 7812, 7956, 8460, 8649, 8892, 9108, 9324, 9540, 10332, 10620, 10764, 10836

Offset: 1

Gus Wiseman, May 15 2022

nonn

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

Said differently, these are nonprime numbers > 1 whose prime shadow is a divisor that is either a prime number or a number already in the sequence.

			The initial terms and their prime indices:
     9: {2,2}
    36: {1,1,2,2}
   125: {3,3,3}
   225: {2,2,3,3}
   441: {2,2,4,4}
  1089: {2,2,5,5}
  1260: {1,1,2,2,3,4}
  1521: {2,2,6,6}
  1980: {1,1,2,2,3,5}

The first term that is not a perfect power A001597 is 1260.
Without the recursion we have A325755 (a superset), counted by A325702.
Before removing the primes we had A353393.
These partitions are counted by A353426 minus one.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A182850 and A323014 give frequency depth, counted by A225485 and A325280.
A325131 lists numbers relatively prime to their prime shadow.
Cf. A000005, A000040, A005117, A316413, A316428, A324850, A353394, A353395, A353397, A353399.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];
Select[Range[2,2000],suQ[#]&&!PrimeQ[#]&]

A353398 Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 2, 1, 2, 1, 2, 6, 5, 4, 4, 6, 6, 8, 8, 13, 16, 13, 16, 18, 16, 20, 21, 27, 30, 27, 33, 41, 44, 51, 48, 58, 61, 66, 66, 74, 83, 86, 99, 102, 111, 115, 126, 137, 147, 156

Offset: 0

Gus Wiseman, May 17 2022

nonn

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The a(8) = 1 through a(14) = 4 partitions (A = 10, B = 11):
  3311  711     61111  521111   5511      B11       A1111
        321111         3221111  9111      721111    731111
                                531111    811111    33221111
                                3321111   5221111   422111111
                                22221111  43111111
                                42111111

The LHS (product of multiplicities) is A005361, counted by A266477.
The RHS (product of prime shadows) is A353394, first appearances A353397.
A related comparison is A353396, ranked by A353395.
These partitions are ranked by A353399.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, counted by A325702.
A339095 counts partitions by product (or factorizations by sum).
Cf. A002033, A003963, A143773, A182850, A239455, A324850, A325756, A353393, A353426.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==Times@@Length/@Split[#]&]],{n,0,30}]

A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 210, 360, 420, 1260, 2310, 2520, 4620, 13860, 27720, 30030, 60060, 75600, 138600, 180180, 360360, 510510, 831600, 900900, 1021020, 1801800, 3063060, 6126120, 9699690, 10810800, 15315300, 19399380, 30630600, 37837800

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

Members m of A025487 such that A181819(m) is also a member of A025487.
If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A181818.
Also the least number with each sorted prime metasignature, where a number's metasignature is the sequence of multiplicities of exponents in its prime factorization. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}. - Gus Wiseman, May 21 2022

			The prime signature of 360360 = 2^3*3^2*5*7*11*13 is (3,2,1,1,1,1). 2 appears as many times as 3 in 360360's prime signature, and 1 appears more times than 2. Since 360360 is also a member of A025487, it is a member of this sequence.
From _Gus Wiseman_, May 21 2022: (Start)
The terms together with their sorted prime signatures and sorted prime metasignatures begin:
      1: {}                -> {}            -> {}
      2: {1}               -> {1}           -> {1}
      6: {1,2}             -> {1,1}         -> {2}
     12: {1,1,2}           -> {1,2}         -> {1,1}
     30: {1,2,3}           -> {1,1,1}       -> {3}
     60: {1,1,2,3}         -> {1,1,2}       -> {1,2}
    210: {1,2,3,4}         -> {1,1,1,1}     -> {4}
    360: {1,1,1,2,2,3}     -> {1,2,3}       -> {1,1,1}
    420: {1,1,2,3,4}       -> {1,1,1,2}     -> {1,3}
   1260: {1,1,2,2,3,4}     -> {1,1,2,2}     -> {2,2}
   2310: {1,2,3,4,5}       -> {1,1,1,1,1}   -> {5}
   2520: {1,1,1,2,2,3,4}   -> {1,1,2,3}     -> {1,1,2}
   4620: {1,1,2,3,4,5}     -> {1,1,1,1,2}   -> {1,4}
  13860: {1,1,2,2,3,4,5}   -> {1,1,1,2,2}   -> {2,3}
  27720: {1,1,1,2,2,3,4,5} -> {1,1,1,2,3}   -> {1,1,3}
  30030: {1,2,3,4,5,6}     -> {1,1,1,1,1,1} -> {6}
  60060: {1,1,2,3,4,5,6}   -> {1,1,1,1,1,2} -> {1,5}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1444 terms from Amiram Eldar)
Eric Weisstein's World of Mathematics, Conjugate Partition.

Intersection of A025487 and A179983.
Subsequence of A129912 and A181826.
Includes all members of A182862.
Positions of first appearances in A353742, unordered version A238747.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A005361 gives product of prime signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A182850 gives frequency depth of prime indices, counted by A225485.
A323014 gives adjusted frequency depth of prime indices, counted by A325280.
Cf. A000040, A055932, A070175, A097318, A304678, A325238, A353507, A353745.

nn=1000;
r=Table[Sort[Length/@Split[Sort[Last/@If[n==1,{},FactorInteger[n]]]]],{n,nn}];
Select[Range[nn],!MemberQ[Take[r,#-1],r[[#]]]&] (* Gus Wiseman, May 21 2022 *)

A318846 Number of balanced reduced multisystems whose atoms cover an initial interval of positive integers with multiplicities equal to the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 4, 15, 11, 20, 21, 90, 51, 80, 32, 468, 166, 2910, 124, 521, 277, 20644, 266, 621, 1761, 1866, 841, 165874, 1374, 1484344, 436, 3797, 12741, 5383, 3108, 14653890, 103783, 31323, 2294, 158136988, 12419, 1852077284, 6382, 20786, 939131, 23394406084

Offset: 1

Gus Wiseman, Sep 04 2018

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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

			The a(12) = 21 multisystems on {1,1,2,3} (commas elided):
  {1123}  {{1}{123}}  {{1}{1}{23}}  {{{1}}{{1}{23}}}
          {{2}{113}}  {{1}{2}{13}}  {{{23}}{{1}{1}}}
          {{3}{112}}  {{1}{3}{12}}  {{{1}}{{2}{13}}}
          {{11}{23}}  {{2}{3}{11}}  {{{2}}{{1}{13}}}
          {{12}{13}}                {{{13}}{{1}{2}}}
                                    {{{1}}{{3}{12}}}
                                    {{{3}}{{1}{12}}}
                                    {{{12}}{{1}{3}}}
                                    {{{2}}{{3}{11}}}
                                    {{{3}}{{2}{11}}}
                                    {{{11}}{{2}{3}}}

Row sums of A330727.
Cf. A001055, A002846, A005121, A181821, A213427, A281118, A281119, A305936, A318762, A318812, A318813.
Cf. A318847, A318848, A318849.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
tmsp[m_]:=Prepend[Join@@Table[tmsp[c],{c,Select[mps[m],1

a(n) = A318812(A181821(n)).
a(prime(n)) = A318813(n).
a(2^n) = A005121(n).

Terminology corrected by Gus Wiseman, Jan 04 2020

More terms from Jinyuan Wang, Jun 26 2020

A332290 Heinz numbers of widely alternately co-strongly normal integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 30, 32, 60, 64, 128, 210, 256, 360, 512, 1024, 2048, 2310, 2520, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288

Offset: 1

Gus Wiseman, Feb 14 2020

An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
This sequence is closed under A181821, so there are infinitely many terms that are not powers of 2 or primorial numbers.

			The sequence of all widely alternately co-strongly normal integer partitions together with their Heinz numbers begins:
      1: ()
      2: (1)
      4: (1,1)
      6: (2,1)
      8: (1,1,1)
     12: (2,1,1)
     16: (1,1,1,1)
     30: (3,2,1)
     32: (1,1,1,1,1)
     60: (3,2,1,1)
     64: (1,1,1,1,1,1)
    128: (1,1,1,1,1,1,1)
    210: (4,3,2,1)
    256: (1,1,1,1,1,1,1,1)
    360: (3,2,2,1,1,1)
    512: (1,1,1,1,1,1,1,1,1)
   1024: (1,1,1,1,1,1,1,1,1,1)
   2048: (1,1,1,1,1,1,1,1,1,1,1)
   2310: (5,4,3,2,1)
   2520: (4,3,2,2,1,1,1)
For example, starting with y = (4,3,2,2,1,1,1), which has Heinz number 2520, and repeatedly taking run-lengths and reversing gives (4,3,2,2,1,1,1) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal with weakly increasing run-lengths and the last is all 1's, so 2520 belongs to the sequence.

Closed under A181821.
The non-co-strong version is A332276.
The enumeration of these partitions by sum is A332289.
The total (rather than alternating) version is A332293.
Cf. A055932, A056239, A100883, A133808, A181819, A317089, A317090, A317246, A317257, A317492, A329747, A332292, A332340.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];
Select[Range[10000],totnQ[Reverse[primeMS[#]]]&]

A367587 Least element in row n of A367858 (multiset multiplicity cokernel).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 1, 6, 4, 3, 1, 7, 1, 8, 1, 4, 5, 9, 1, 3, 6, 2, 1, 10, 3, 11, 1, 5, 7, 4, 2, 12, 8, 6, 1, 13, 4, 14, 1, 2, 9, 15, 1, 4, 1, 7, 1, 16, 1, 5, 1, 8, 10, 17, 1, 18, 11, 2, 1, 6, 5, 19, 1, 9, 4, 20, 1, 21, 12, 2, 1, 5, 6, 22, 1, 2

Offset: 1

Gus Wiseman, Dec 03 2023

nonn

We define the multiset multiplicity cokernel MMC(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}. As an operation on multisets MMC is represented by A367858, and as an operation on their ranks it is represented by A367859.

Indices of first appearances are A008578.
Depends only on rootless base A052410, see A007916.
For kernel instead of cokernel we have A055396.
For maximum instead of minimum element we have A061395.
The opposite version is A367583.
Row-minima of A367858.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 lists prime multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, sorted A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A367579 lists MMK, rank A367580, sum A367581, max A367583, min A055396.
Cf. A000720, A027746, A051904, A071625, A072774, A130091, A367582, A367584, A367585, A367586.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&],{i,mts}]]];
Table[If[n==1,0,Min@@mmc[prix[n]]],{n,100}]

a(n) = A055396(A367859(n)).
a(n^k) = a(n) for all positive integers n and k.
If n is a power of a squarefree number, a(n) = A061395(n).

A318287 Number of non-isomorphic strict multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 3, 4, 5, 3, 7, 4, 7, 9, 5, 5, 12, 6, 12, 14, 10, 8, 13, 12, 14, 14, 18, 10, 34

Offset: 1

Gus Wiseman, Aug 23 2018

			Non-isomorphic representatives of the a(20) = 12 strict multiset partitions of {1,1,1,2,3}:
  {{1,1,1,2,3}}
  {{1},{1,1,2,3}}
  {{2},{1,1,1,3}}
  {{1,1},{1,2,3}}
  {{1,2},{1,1,3}}
  {{2,3},{1,1,1}}
  {{1},{2},{1,1,3}}
  {{1},{1,1},{2,3}}
  {{1},{1,2},{1,3}}
  {{2},{3},{1,1,1}}
  {{2},{1,1},{1,3}}
  {{1},{2},{3},{1,1}}

Cf. A001055, A007716, A045778, A181821, A305936, A316980, A317533, A317776, A317791.

Cf. A318283, A318284, A318285, A318286, A318357, A318362, A318371.

a(n) = A318357(A181821(n)).

A318362 Number of non-isomorphic set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 3, 2, 1, 5, 1, 2, 3, 5, 1, 7, 1, 5, 3, 2, 1, 9, 4, 2, 8, 5, 1, 10

Offset: 1

Gus Wiseman, Aug 24 2018

			Non-isomorphic representatives of the a(12) = 5 set multipartitions of {1,1,2,3}:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{1},{2},{3}}

Cf. A007716, A049311, A116540, A181821, A317791.

Cf. A318283, A318285, A318287, A318360, A318361, A318369, A318371.

a(n) = A318369(A181821(n)).

A318848 Number of complete tree-partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 4, 12, 9, 12, 17, 34, 29, 44, 26, 92, 90, 277, 68, 171, 93, 806, 144, 197, 309, 581, 269, 2500, 428, 7578, 236, 631, 1025, 869, 954, 24198, 3463, 2402, 712, 75370, 1957, 243800, 1040, 3200, 11705, 776494, 1612, 4349, 2358, 8862, 3993, 2545777

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A tree-partition of m is either m itself or a sequence of tree-partitions, one of each part of a multiset partition of m with at least two parts. A tree-partition is complete if the leaves are all multisets of length 1.

			The a(12) = 17 complete tree-partitions of {1,1,2,3} with the leaves (x) replaced with just x:
  (1(1(23)))
  (1(2(13)))
  (1(3(12)))
  (2(1(13)))
  (2(3(11)))
  (3(1(12)))
  (3(2(11)))
  ((11)(23))
  ((12)(13))
  (1(123))
  (2(113))
  (3(112))
  (11(23))
  (12(13))
  (13(12))
  (23(11))
  (1123)

Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318847, A318849.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
allmsptrees[m_]:=Prepend[Join@@Table[Tuples[allmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Length[Select[allmsptrees[nrmptn[n]],FreeQ[#,{?AtomQ,_}]&]],{n,20}]

a(n) = A281119(A181821(n)).
a(prime(n)) = A196545(n)
a(2^n) = A000311(n).

More terms from Jinyuan Wang, Jun 26 2020

A329600 Smallest number with the same set of distinct prime exponents as A108951(n).

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 8, 4, 12, 2, 24, 2, 12, 12, 16, 2, 72, 2, 24, 12, 12, 2, 48, 4, 12, 8, 24, 2, 360, 2, 32, 12, 12, 12, 144, 2, 12, 12, 48, 2, 360, 2, 24, 24, 12, 2, 96, 4, 72, 12, 24, 2, 432, 12, 48, 12, 12, 2, 720, 2, 12, 24, 64, 12, 360, 2, 24, 12, 360, 2, 288, 2, 12, 72, 24, 12, 360, 2, 96, 16, 12, 2, 720, 12, 12, 12, 48, 2, 2160, 12, 24, 12, 12, 12

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A007947, A034386, A071364, A108951, A122111, A181819, A181821, A328400, A329607.

Cf. A077462 (rgs-transform, from its term a(1)=1 onward).

Array[Times @@ MapIndexed[Prime[#2[[1]]]^#1 &, Reverse[Flatten[Cases[FactorInteger[#], {p_, k_} :> Table[PrimePi[p], {k}]]]]] &[Times @@ FactorInteger[#][[All, 1]]] &@ If[# == 1, 1, Times @@ Prime@ FactorInteger[#][[All, -1]]] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105] (* Michael De Vlieger, Nov 18 2019, after Gus Wiseman at A181821 *)

A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
A328400(n) = A181821(A007947(A181819(n)));
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329600(n) = A328400(A108951(n));

a(n) = A328400(A108951(n)) = A181821(A007947(A122111(n)))

a(n) = A181821(A329607(n)) = A181821(A122111(A071364(n))).

cp's OEIS Frontend

A353389 Create the sequence of all positive integers > 1 that are prime or whose prime shadow (A181819) is a divisor that is already in the sequence. Then remove all the primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A353398 Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A318846 Number of balanced reduced multisystems whose atoms cover an initial interval of positive integers with multiplicities equal to the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A332290 Heinz numbers of widely alternately co-strongly normal integer partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A367587 Least element in row n of A367858 (multiset multiplicity cokernel).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A318287 Number of non-isomorphic strict multiset partitions of a multiset whose multiplicities are the prime indices of n.

Views

Author

Keywords

Examples

Crossrefs

Formula

A318362 Number of non-isomorphic set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

Views

Author

Keywords

Examples

Crossrefs

Formula

A318848 Number of complete tree-partitions of a multiset whose multiplicities are the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs