A181821 -id:A181821 - cp's OEIS frontend

A316656 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

0, 1, 0, 1, 0, 1, 0, 4, 3, 1, 0, 9, 0, 1, 6, 26, 0, 36, 0, 16, 10, 1, 0, 92, 21, 1, 197, 25, 0, 100, 0, 236, 15, 1, 53, 474

Offset: 1

Gus Wiseman, Jul 09 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of sets of trees begins:
   1:
   2: 1
   3:
   4: (12)
   5:
   6: (1(12))
   7:
   8: (1(23)), (2(13)), (3(12)), (123)
   9: (1(2(12))), (2(1(12))), (12(12))
  10: (1(1(12)))
  11:
  12: (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), ((12)(13)), (12(13)), (13(12))

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.

Cf. A316651, A316652, A316653. A316654, A316655.

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]]]];
gro[m_]:=If[Length[m]==1,m,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
Table[Length[gro[Flatten[MapIndexed[Table[#2,{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]],{n,30}]

a(prime(n>1)) = 0.

a(2^n) = A000311(n).

A332672 Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 6, 0, 0, 6, 16, 0, 21, 0, 12, 10, 0, 0, 48, 16, 0, 81, 20, 0, 48, 0, 104, 15, 0, 30, 162, 0, 0, 21, 104, 0, 90, 0, 30, 198, 0, 0, 336, 65, 124, 28, 42, 0, 603, 50, 190, 36, 0, 0, 396, 0, 0, 405, 688, 77, 150, 0, 56, 45, 260, 0

Offset: 1

Gus Wiseman, Feb 23 2020

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 sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(n) permutations for n = 8, 9, 12, 15, 16:
  213   1212   1213   11212   1324
  312   2112   1312   12112   1423
        2121   2113   12121   2134
               2131   21112   2143
               3112   21121   2314
               3121   21211   2413
                              3124
                              3142
                              3214
                              3241
                              3412
                              4123
                              4132
                              4213
                              4231
                              4312

MathWorld, Unimodal Sequence

Positions of zeros are one and A001751.
Support is A264828 without one.
Dominated by A318762.
The complement is counted by A332294.
A less interesting version is A332671.
The opposite version is A332742.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Compositions whose negation is not unimodal are A332669.
Cf. A007052, A008480, A056239, A112798, A124010, A181819, A181821, A332281, A332287, A332294, A332642, A332741.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]],!unimodQ[#]&]],{n,30}]

a(n) = A332671(A181821(n)).

a(n) + A332294(n) = A318762(n).

A332741 Number of unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 8, 4, 2, 9, 4, 1, 6, 1, 16, 3, 2, 4, 12, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 16, 5, 8, 3, 4, 1, 18, 4, 8, 3, 2, 1, 12, 1, 2, 9, 32, 4, 6, 1, 4, 3, 8, 1, 24, 1, 2, 12, 4, 5, 6, 1, 16, 27, 2, 1

Offset: 1

Gus Wiseman, Mar 09 2020

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 sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(12) = 4 permutations:
  {1,1,2,3}
  {2,1,1,3}
  {3,1,1,2}
  {3,2,1,1}

Eric Weisstein's World of Mathematics, Unimodal Sequence

Dominated by A318762.
The non-negated version is A332294.
The complement is counted by A332742.
A less interesting version is A333145.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Numbers with non-unimodal negated prime signature are A332642.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Partitions with unimodal negated run-lengths are A332638.
Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332288, A332639, A332669, A332672.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]],unimodQ[-#]&]],{n,30}]

a(n) + A332742(n) = A318762(n).

A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1152, 1296, 1728, 2048, 2592, 3456, 5184, 7776, 8192, 10368, 13824, 15552, 18432, 20736, 31104, 41472, 55296, 62208, 73728, 86400, 108000, 129600, 131072, 165888, 194400, 216000, 221184, 259200, 279936, 324000

Offset: 1

Gus Wiseman, May 17 2022

nonn

All terms are highly powerful (A005934), but that sequence looks only at first appearances that reach a record, and is missing 1152, 2048, 8192, etc.

			The prime exponents of 86400 are (7,3,2), and this is the first case of product 42, so 86400 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Highly powerful number.

These are the positions of first appearances in A005361, counted by A266477.
This is the sorted version of A085629.
The version for shadows instead of exponents is A353397, firsts in A353394.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices, counted by A339095.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime exponents, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
Cf. A070175, A097318, A116608, A162247, A182850, A304678, A325131, A325238, A353399, A353503, A353506, A353507.
Subsequence of A181800.

nn=1000;
d=Table[Times@@Last/@FactorInteger[n],{n,nn}];
Select[Range[nn],!MemberQ[Take[d,#-1],d[[#]]]&]
lps[fct_] := Module[{nf = Length[fct]}, Times @@ (Prime[Range[nf]]^Reverse[fct])]; lps[{1}] = 1; q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, (n == 1 || AllTrue[e, # > 1 &]) && n == Min[lps /@ f[Times @@ e]]]; Select[Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]], q] (* Amiram Eldar, Sep 29 2024, using the function f by T. D. Noe at A162247 *)

A382858 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all equal.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 4, 0, 1, 6, 1, 0, 1, 24, 1, 12, 1, 2, 1, 0, 1, 36, 4, 0, 36, 0, 1, 10, 1, 120, 0, 0, 1, 84, 1, 0, 0, 24, 1, 3, 1, 0, 38, 0, 1, 240, 6, 18, 0, 0, 1, 246, 0, 6, 0, 0, 1, 96, 1, 0, 30, 720, 1, 0, 1, 0, 0, 14, 1, 660, 1, 0, 74, 0, 1, 0, 1

Offset: 1

Gus Wiseman, Apr 09 2025

nonn

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). 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}.

			The a(9) = 4 permutations are:
  (1,1,2,2)
  (1,2,1,2)
  (2,1,2,1)
  (2,2,1,1)

The anti-run case is A335125.
These permutations for factorials are counted by A335407, distinct A382774.
For distinct instead of equal run-lengths we have A382773.
For prime indices we have A382857 (firsts A382878), distinct A382771 (firsts A382772).
Positions of 0 are A382914, signature restriction of A382915.
A003963 gives product of prime indices.
A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.
A047966 counts partitions with equal multiplicities, distinct A098859.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A382913 ranks Look-and-Say partitions by signature, complement A382912.
Cf. A000720, A000961, A001221, A001222, A003242, A048767, A181821, A182854, A238130, A305936, A351202, A382879.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Length[Select[Permutations[nrmptn[n]],SameQ@@Length/@Split[#]&]],{n,100}]

a(n) = A382857(A181821(n)) = A382857(A304660(n)).

A050322 Number of factorizations indexed by prime signatures: A001055(A025487).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 5, 7, 9, 12, 11, 11, 16, 19, 21, 15, 29, 26, 30, 15, 31, 38, 22, 47, 52, 45, 36, 57, 64, 30, 77, 98, 67, 74, 97, 66, 105, 42, 109, 118, 92, 109, 171, 97, 141, 162, 137, 165, 56, 212, 181, 52, 198, 189, 289, 139, 250, 257, 269, 254, 77, 382, 267

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

For A025487(m) = 2^k = A000079(k), we have a(m) = A000041(k).

Is a(k) = A000110(k) for A025487(m) = A002110(k)?

			From _Gus Wiseman_, Jan 13 2020: (Start)
The a(1) = 1 through a(11) = 9 factorizations:
  {}  2  4    6    8      12     16       24       30     32         36
         2*2  2*3  2*4    2*6    2*8      3*8      5*6    4*8        4*9
                   2*2*2  3*4    4*4      4*6      2*15   2*16       6*6
                          2*2*3  2*2*4    2*12     3*10   2*2*8      2*18
                                 2*2*2*2  2*2*6    2*3*5  2*4*4      3*12
                                          2*3*4           2*2*2*4    2*2*9
                                          2*2*2*3         2*2*2*2*2  2*3*6
                                                                     3*3*4
                                                                     2*2*3*3
(End)

R. J. Mathar and Michael De Vlieger, Table of n, a(n) for n = 1..5000 (First 300 terms from _R. J. Mathar_)
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

Cf. A000041, A000079, A000110, A001055, A002110, A025487.
The version indexed by unsorted prime signature is A331049.
The version indexed by prime shadow (A181819, A181821) is A318284.
This sequence has range A045782 (same as A001055).
Cf. A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

A050322 := proc(n)
    A001055(A025487(n)) ;
end proc: # R. J. Mathar, May 25 2017

c[1, r_] := c[1, r] = 1; c[n_, r_] := c[n, r] = Module[{d, i}, d = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/d[[i]], d[[i]]], {i, 1, Length[d]}]]; Map[c[#, #] &, Union@ Table[Times @@ MapIndexed[If[n == 1, 1, Prime[First@ #2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, Product[Prime@ i, {i, 6}]}]] (* Michael De Vlieger, Jul 10 2017, after Dean Hickerson at A001055 *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Length/@facs/@First/@GatherBy[Range[1000],If[#==1,{},Sort[Last/@FactorInteger[#]]]&] (* Gus Wiseman, Jan 13 2020 *)

A316654 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 1, 5, 39, 387, 4960, 74088, 1312716, 26239484, 595023510, 14908285892, 412903136867, 12448252189622, 407804188400373, 14380454869464352, 544428684832123828, 21991444994187529639, 945234507638271696504, 43042162953650721470752, 2071216980365429970912347

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.

			The a(3) = 5 trees are (1(12)), (1(23)), (2(13)), (3(12)), (123).

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A141268, A181821, A292504, A300660, A304660.

Cf. A316651, A316652, A316653, A316655, A316656.

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]]]];
gro[m_]:=If[Length[m]==1,m,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n]=polcoef(sWeighT(x*Ser(v[1..n])), n)); x*Ser(v)}
StronglyNormalLabelingsSeq(cycleIndexSeries(12)) \\ Andrew Howroyd, Jan 22 2021

Terms a(9) and beyond from Andrew Howroyd, Jan 22 2021

A318849 Number of orderless tree-partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 11, 8, 27, 20, 30, 38, 96, 74, 114, 58, 308, 234, 1052, 176, 509, 278, 3648, 374, 600, 1076, 1760, 814, 13003, 1306, 47006, 612, 2226, 4200, 3094, 2914, 172605, 16588, 9814, 2168, 640662, 6998, 2402388, 3698, 11496, 65936, 9082538, 4914, 17996

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 multiset of tree-partitions, one of each part of a multiset partition of m with at least two parts.

			The a(7) = 11 orderless tree-partitions of {1,1,1,1}:
  (1111)
  ((1)(111))
  ((11)(11))
  ((1)(1)(11))
  ((1)((1)(11)))
  ((11)((1)(1)))
  ((1)(1)(1)(1))
  ((1)((1)(1)(1)))
  ((1)(1)((1)(1)))
  ((1)((1)((1)(1))))
  (((1)(1))((1)(1)))

Cf. A000311, A001055, A196545, A292504, A292505, A305936, A316655, A318762, A318812, A318813, A318847.

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}]]]]];
olmsptrees[m_]:=Prepend[Union@@Table[Sort/@Tuples[olmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Length[olmsptrees[nrmptn[n]]],{n,15}]

a(n) = A292504(A181821(n)).
a(prime(n)) = A141268(n).
a(2^n) = A005804(n).

More terms from Jinyuan Wang, Jun 26 2020

A328400 Smallest number with the same set of distinct prime exponents as n.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 8, 4, 2, 2, 12, 2, 2, 2, 16, 2, 12, 2, 12, 2, 2, 2, 24, 4, 2, 8, 12, 2, 2, 2, 32, 2, 2, 2, 4, 2, 2, 2, 24, 2, 2, 2, 12, 12, 2, 2, 48, 4, 12, 2, 12, 2, 24, 2, 24, 2, 2, 2, 12, 2, 2, 12, 64, 2, 2, 2, 12, 2, 2, 2, 72, 2, 2, 12, 12, 2, 2, 2, 48, 16, 2, 2, 12, 2, 2, 2, 24, 2, 12, 2, 12, 2, 2, 2, 96, 2, 12, 12, 4, 2, 2, 2, 24, 2

Offset: 1

Antti Karttunen, Oct 15 2019

nonn

A variant of A046523 which gives the smallest number with the same prime signature as n. However, in this sequence, if any prime exponent occurs multiple times in n, the extra occurrences are removed and the signature is that of one of the numbers where only distinct values of prime exponents occur (A130091).

			90 = 2^1 * 3^2 * 5^1 has prime signature (1,1,2). The smallest number with prime signature (1,2) is 12 = 2^2 * 3, thus a(90) = 12.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A007947, A046523, A181819, A181821, A328401 (rgs-transform).

Cf. A005117 (gives indices of terms <= 2), A062503 (after its initial 1, gives indices of 4's in this sequence).

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]]] &, 105] (* Michael De Vlieger, Oct 17 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)));

a(n) = A181821(A007947(A181819(n))).

For all n, a(n) = a(A046523(n)).

A328835 Prime shadow of primorial base exp-function: a(n) = A181819(A276086(n)).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 2, 4, 4, 8, 6, 12, 3, 6, 6, 12, 9, 18, 5, 10, 10, 20, 15, 30, 7, 14, 14, 28, 21, 42, 2, 4, 4, 8, 6, 12, 4, 8, 8, 16, 12, 24, 6, 12, 12, 24, 18, 36, 10, 20, 20, 40, 30, 60, 14, 28, 28, 56, 42, 84, 3, 6, 6, 12, 9, 18, 6, 12, 12, 24, 18, 36, 9, 18, 18, 36, 27, 54, 15, 30, 30, 60, 45, 90, 21, 42, 42, 84, 63, 126, 5, 10, 10, 20, 15, 30, 10, 20, 20

Offset: 0

Antti Karttunen, Oct 29 2019

nonn

From Antti Karttunen, Apr 30 2022: (Start)
These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of times a nonzero digit k occurs in the primorial base representation of n.
Note that this sequence, and all the sequences derived from it as b(n) = f(a(n)), [where f is any integer-valued function] can be represented as b(n) = g(A278226(n)), where g(n) = f(A181819(n)). E.g., if f is the identity function (so that b(n) is this sequence), then g(n) is A181819(n). See the comment and formulas in the latter sequence.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial numbers

Cf. A001222, A007814, A061395, A181819, A181821, A267263, A276086, A278226 (A286626), A328114, A328614.

Cf. A275735 for analogous sequence.

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328835(n) = A181819(A276086(n));

a(n) = A181819(A276086(n)).
A001222(a(n)) = A267263(n).
A007814(a(n)) = A328614(n).
A061395(a(n)) = A328114(n).
For all n >= 0, a(n) = A181819(A278226(n)) and A181821(a(n)) = A278226(n). - Antti Karttunen, Apr 30 2022

cp's OEIS Frontend

A316656 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

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A332672 Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n.

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Mathematica

Formula

A332741 Number of unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

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Mathematica

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A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

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Mathematica

A382858 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all equal.

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Mathematica

Formula

A050322 Number of factorizations indexed by prime signatures: A001055(A025487).

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Maple

Mathematica

A316654 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

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Mathematica

PARI

Extensions

A318849 Number of orderless tree-partitions of a multiset whose multiplicities are the prime indices of n.

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