A008480 -id:A008480 - cp's OEIS frontend

A034729 a(n) = Sum_{ k, k|n } 2^(k-1).

1, 3, 5, 11, 17, 39, 65, 139, 261, 531, 1025, 2095, 4097, 8259, 16405, 32907, 65537, 131367, 262145, 524827, 1048645, 2098179, 4194305, 8390831, 16777233, 33558531, 67109125, 134225995, 268435457, 536887863, 1073741825, 2147516555, 4294968325, 8590000131

Offset: 1

Erich Friedman

nonn

Dirichlet convolution of b_n=1 with c_n = 2^(n-1).
Equals row sums of triangle A143425, & inverse Möbius transform (A051731) of [1, 2, 4, 8, ...]. - Gary W. Adamson, Aug 14 2008
Number of constant multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The a(4) = 11 constant multiset partitions:
  (1)(1)(1)(1)
    (11)(11)
    (12)(12)
     (1111)
     (1222)
     (1122)
     (1112)
     (1233)
     (1223)
     (1123)
     (1234)
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000005, A002033, A003238, A008480, A018783, A047968, A051731.
Cf. A052409, A055895, A078392, A143425, A215366, A245282, A248906.
Cf. A289508.
Sums of the form Sum_{d|n} q^(d-1): this sequence (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A034729:= func< n | (&+[2^(d-1): d in Divisors(n)]) >;
[A034729(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

seq(add(2^(k-1),k=numtheory:-divisors(n)), n = 1 .. 100); # Robert Israel, Aug 22 2014

Rest[CoefficientList[Series[Sum[x^k/(1-2*x^k),{k,1,30}],{x,0,30}],x]] (* Vaclav Kotesovec, Sep 08 2014 *)

A034729(n) = sumdiv(n,k,2^(k-1)) \\ Michael B. Porter, Mar 11 2010

{a(n)=polcoeff(sum(m=1,n,2^(m-1)*x^m/(1-x^m +x*O(x^n))),n)}
for(n=1,40,print1(a(n),", ")) \\ Paul D. Hanna, Aug 21 2014

{a(n)=local(A=x+x^2);A=sum(m=1,n,x^m*sumdiv(m,d,1/(1 - x^(m/d) +x*O(x^n))^d) );polcoeff(A,n)}
for(n=1,40,print1(a(n),", ")) \\ Paul D. Hanna, Aug 21 2014

from sympy import divisors
def A034729(n): return sum(1<<(d-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jul 15 2022

def A034729(n): return sum(2^(k-1) for k in (1..n) if (k).divides(n))
[A034729(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

G.f.: Sum_{n>0} x^n/(1-2*x^n). - Vladeta Jovovic, Nov 14 2002
a(n) = 1/2 * A055895(n). - Joerg Arndt, Aug 14 2012
G.f.: Sum_{n>=1} 2^(n-1) * x^n / (1 - x^n). - Paul D. Hanna, Aug 21 2014
G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1 - x^d)^(n/d). - Paul D. Hanna, Aug 21 2014
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 09 2014
a(n) = Sum_{k in row n of A215366} A008480(k) * A000005(A289508(k)). - Gus Wiseman, Sep 16 2018
a(n) = Sum_{c is a composition of n} A000005(gcd(c)). - Gus Wiseman, Sep 16 2018

A319193 Irregular triangle where T(n,k) is the number of permutations of the integer partition with Heinz number A215366(n,k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 2, 1, 1, 3, 6, 6, 4, 5, 1, 1, 2, 2, 2, 6, 3, 3, 3, 4, 4, 12, 10, 5, 6, 1, 1, 2, 2, 1, 3, 2, 3, 6, 6, 3, 1, 12, 4, 12, 6, 10, 5, 20, 15, 6, 7, 1, 1, 2, 2, 2, 3, 2, 6, 3, 3, 4, 6, 6, 1, 12, 12, 4, 12

Offset: 0

Gus Wiseman, Sep 13 2018

A refinement of Pascal's triangle, these are the unsigned coefficients appearing in the expansion of homogeneous symmetric functions in terms of elementary symmetric functions.

			Triangle begins:
  1
  1
  1  1
  1  2  1
  1  1  2  3  1
  1  2  2  3  3  4  1
  1  2  2  1  1  3  6  6  4  5  1
The fourth row corresponds to the symmetric function identity: h(4) = -e(4) + e(22) + 2 e(31) - 3 e(211) + e(1111).

Alois P. Heinz, Rows n = 0..33, flattened

A different row ordering is A072811.

Cf. A000041, A005651, A008277, A008480, A056239, A124794, A215366, A319182, A319191, A319192.

b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n], [seq(
      map(p-> p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
    end:
T:= n-> map(m-> (l-> add(i, i=l)!/mul(i!, i=l))(map(
        i-> i[2], ifactors(m)[2])), sort(b(n$2)))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 14 2020

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Permutations[primeMS[k]]],{n,6},{k,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0 || i < 2, {2^n}, Flatten[Table[ #*Prime[i]^j& /@ b[n - i*j, i - 1], {j, 0, n/i}]]];
T[n_] := Map[Function[m, Function[l, Total[l]!/Times @@ (l!)][ FactorInteger[m][[All, 2]]]], Sort[b[n, n]]];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

T(n,k) = A008480(A215366(n,k)).

T(0,1)=1 prepended by Alois P. Heinz, Feb 14 2020

A345164 Number of alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 4, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 4, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A335452 at a(30) = 4, A335452(30) = 6. The anti-runs (2,3,5) and (5,3,2) are not alternating.

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutation, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The a(n) alternating permutations of prime indices for n = 180, 210, 300, 420, 900:
  (12132)  (1324)  (13132)  (12143)  (121323)
  (21213)  (1423)  (13231)  (13142)  (132312)
  (21312)  (2143)  (21313)  (13241)  (213132)
  (23121)  (2314)  (23131)  (14132)  (213231)
  (31212)  (2413)  (31213)  (14231)  (231213)
           (3142)  (31312)  (21314)  (231312)
           (3241)           (21413)  (312132)
           (3412)           (23141)  (323121)
           (4132)           (24131)
           (4231)           (31214)
                            (31412)
                            (34121)
                            (41213)
                            (41312)

Counting all permutations gives A008480.
Dominated by A335452 (number of separations of prime factors).
Including twins (x,x) gives A344606.
Positions of zeros are A345171, counted by A345165.
Positions of nonzero terms are A345172.
A000041 counts integer partitions.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344654 counts non-twin partitions w/o alternating permutation, rank: A344653.
A344740 counts twins and partitions w/ alternating permutation, rank: A344742.
A345166 counts separable partitions w/o alternating permutation, rank: A345173.
A345170 counts partitions with a alternating permutation.
Cf. A001222, A071321, A071322, A316523, A316524, A333489, A335126, A344605, A344614, A344616, A344652, A345163, A345168.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],wigQ]],{n,30}]

A321742 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in e(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 3, 0, 0, 0, 0, 1, 1, 3, 6, 0, 1, 0, 2, 6, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 5, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 5, 0, 0, 0, 1, 0, 3, 10, 1, 6, 4, 12, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 19 2018

Row n has length A000041(A056239(n)).

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

			Triangle begins:
   1
   1
   0   1
   1   2
   0   0   1
   0   1   3
   0   0   0   0   1
   1   3   6
   0   1   0   2   6
   0   0   0   1   4
   0   0   0   0   0   0   1
   0   2   1   5  12
   0   0   0   0   0   0   0   0   0   0   1
   0   0   0   0   0   1   5
   0   0   0   1   0   3  10
   1   6   4  12  24
   0   0   0   0   0   0   0   0   0   0   0   0   0   0   1
   0   0   1   5   2  12  30
For example, row 12 gives: e(211) = 2m(22) + m(31) + 5m(211) + 12m(1111).

Wikipedia, Symmetric polynomial

Row sums are A321743.

Cf. A008480, A049311, A056239, A116540, A124794, A124795, A300121, A319193, A321738, A321742-A321765, A321854.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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}]]]]];
Table[Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,Select[mps[nrmptn[n]],And[And@@UnsameQ@@@#,Sort[Length/@#]==primeMS[k]]&]}],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}],{n,18}]

A321765 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 0, -1, 1, 0, -1, 1, -1, 1, 2, 1, 1, 2, -1, -1, 1, 1, -1, 0, 0, 1, 1, -1, 0, 0, 1, -1, 1, 1, 0, 1, -1, -1, 1, 0, -1, 0, 0, 1, 0, 0, -1, 1, -1, 1, 0, -1, 1, 0, 0, -1

Offset: 1

Gus Wiseman, Nov 20 2018

Row n has length A000041(A056239(n)).

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

			Triangle begins:
   1
   1
   1  -1
   1   1
   1  -1   1
   1   0  -1
   1   0  -1   1  -1
   1   2   1
   1   2  -1  -1   1
   1  -1   0   0   1
   1  -1   0   0   1  -1   1
   1   0   1  -1  -1
   1   0  -1   0   0   1   0   0  -1   1  -1
   1   0  -1   1   0   0  -1
For example, row 12 gives: p(211) = s(4) + s(31) - s(211) - s(1111).

Wikipedia, Symmetric polynomial

Row sums are A317554.

Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A296561, A300121, A304438, A317552, A317554, A321742-A321765.

A351202 Number of permutations of the multiset of prime factors of n (or ordered prime factorizations of n) with all distinct runs.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 6, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 2, 1, 2, 6, 1, 2, 2, 6, 1, 4, 1, 2, 2, 2, 2, 6, 1, 4, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 13 2022

nonn

			The a(36) = 2 permutations are (1,1,2,2), (2,2,1,1). Missing are: (1,2,1,2), (1,2,2,1), (2,1,1,2), (2,1,2,1). Here we use prime indices instead of factors.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The maximum number of possible permutations is A008480.
Positions less than A008480 are A351201.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A283353 counts normal multisets with a permutation without distinct runs.
A297770 counts distinct runs in binary expansion.
A351014 counts distinct runs in standard compositions, firsts A351015.
A351204 = partitions whose perms. have distinct runs, complement A351203.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
Cf. A001055, A001221, A001222, A003242, A061395, A116608, A238130 or A238279, A328592, A351291.

Table[Length[Select[Permutations[Join@@ ConstantArray@@@FactorInteger[n]],UnsameQ@@Split[#]&]],{n,100}]

A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!1!2!...n!).

Original entry on oeis.org

1, 1, 3, 60, 12600, 37837800, 2053230379200, 2431106898187968000, 73566121315513295589120000, 65191584694745586153436251091200000, 1906765806522767212441719098019963758016000000, 2048024348726152339387799085049745725891853852479488000000

Offset: 0

Clark Kimberling

Number of ways to put numbers 1, 2, ..., n*(n+1)/2 in a triangular array of n rows in such a way that each row is increasing. Also number of ways to choose groups of 1, 2, 3, ..., n-1 and n objects out of n*(n+1)/2 objects. - Floor van Lamoen, Jul 16 2001
a(n) is the number of ways to linearly order the multiset {1,2,2,3,3,3,...n,n,...n}. - Geoffrey Critzer, Mar 08 2009
Also the number of distinct adjacency matrices in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017

			From _Gus Wiseman_, Aug 12 2020: (Start)
The a(3) = 60 permutations of the prime indices of A006939(3) = 360:
  (111223)  (121123)  (131122)  (212113)  (231211)
  (111232)  (121132)  (131212)  (212131)  (232111)
  (111322)  (121213)  (131221)  (212311)  (311122)
  (112123)  (121231)  (132112)  (213112)  (311212)
  (112132)  (121312)  (132121)  (213121)  (311221)
  (112213)  (121321)  (132211)  (213211)  (312112)
  (112231)  (122113)  (211123)  (221113)  (312121)
  (112312)  (122131)  (211132)  (221131)  (312211)
  (112321)  (122311)  (211213)  (221311)  (321112)
  (113122)  (123112)  (211231)  (223111)  (321121)
  (113212)  (123121)  (211312)  (231112)  (321211)
  (113221)  (123211)  (211321)  (231121)  (322111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..35
Eric Weisstein's World of Mathematics, Adjacency Matrix
Eric Weisstein's World of Mathematics, Multinomial Coefficient

Cf. A000178, A014068, A022919, A052295, A208437, A327803.
A190945 counts the case of anti-run permutations.
A317829 counts partitions of this multiset.
A325617 is the version for factorials instead of superprimorials.
A006939 lists superprimorials or Chernoff numbers.
A008480 counts permutations of prime indices.
A181818 gives products of superprimorials, with complement A336426.
Cf. A000142, A022559, A027423, A034841, A076954, A112798, A303279, A336417.

with(combinat):
a:= n-> multinomial(binomial(n+1, 2), $0..n):
seq(a(n), n=0..12);  # Alois P. Heinz, May 18 2013

Table[Apply[Multinomial ,Range[n]], {n, 0, 20}]  (* Geoffrey Critzer, Dec 09 2012 *)
Table[Multinomial @@ Range[n], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
Table[Binomial[n + 1, 2]!/BarnesG[n + 2], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
Table[Length[Permutations[Join@@Table[i,{i,n},{i}]]],{n,0,4}] (* Gus Wiseman, Aug 12 2020 *)

a(n) = binomial(n+1,2)!/prod(k=1, n, k^(n+1-k)); \\ Michel Marcus, May 02 2019

a(n) = (n*(n+1)/2)!/(0!*1!*2!*...*n!).
a(n) = a(n-1) * A014068(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001.
a(n) = A052295(n)/A000178(n). - Lekraj Beedassy, Feb 19 2004
a(n) = A208437(n*(n+1)/2,n). - Alois P. Heinz, Apr 08 2016
a(n) ~ A * exp(n^2/4 + n + 1/6) * n^(n^2/2 + 7/12) / (2^((n+1)^2/2) * Pi^(n/2)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 02 2019
a(n) = A327803(n*(n+1)/2,n). - Alois P. Heinz, Sep 25 2019
a(n) = A008480(A006939(n)). - Gus Wiseman, Aug 12 2020

More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2001
More terms from Michel ten Voorde, Apr 12 2001
Better definition from L. Edson Jeffery, May 18 2013

A319191 Coefficient of p(y) / A056239(n)! in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, -1, 1, 2, -3, -6, 1, 3, 8, 24, -6, -120, -30, -20, 1, 720, 15, -5040, 20, 90, 144, 40320, -10, 40, -840, -15, -90, -362880, -120, 3628800, 1, -504, 5760, -420, 45, -39916800, -45360, 3360, 40, 479001600, 630, -6227020800, 504, 210, 403200, 87178291200

Offset: 1

Gus Wiseman, Sep 13 2018

sign

A refinement of Stirling numbers of the first kind.

An unsigned version is A124795.

Cf. A000041, A000110, A000258, A005651, A008480, A048994, A056239, A124794, A215366, A318762, A319182.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[(-1)^(Total[primeMS[n]]-PrimeOmega[n])*numPermsOfType[primeMS[n]],{n,100}]

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) = (-1)^(Sum x_i * y_i - Sum y_i) (Sum x_i * y_i)! / (Product x_i^y_i * Product y_i!).

A318992 Numbers whose consecutive prime indices are not all divisible.

Original entry on oeis.org

15, 30, 33, 35, 45, 51, 55, 60, 66, 69, 70, 75, 77, 85, 90, 91, 93, 95, 99, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 150, 153, 154, 155, 161, 165, 170, 175, 177, 180, 182, 186, 187, 190, 195, 198, 201, 203, 204, 205, 207, 209, 210, 215

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The sequence of partitions whose Heinz numbers belong to the sequence begins: (3,2), (3,2,1), (5,2), (4,3), (3,2,2), (7,2), (5,3), (3,2,1,1), (5,2,1), (9,2), (4,3,1), (3,3,2), (5,4), (7,3), (3,2,2,1), (6,4), (11,2), (8,3), (5,2,2).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000040, A001221, A001222, A008480, A289509, A300912, A318990, A318991, A318993.

Select[Range[100],!Or[#==1,PrimePowerQ[#],Divisible@@Reverse[PrimePi/@FactorInteger[#][[All,1]]]]&]

ok(n)={my(v=apply(primepi, factor(n)[,1])); for(i=2, #v, if(v[i]%v[i-1], return(1))); 0} \\ Andrew Howroyd, Oct 26 2018

A382857 Number of ways to permute the prime indices of n so that the run-lengths are all equal.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 2, 4, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 1, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 1, 2, 1, 6, 2, 2

Offset: 0

Gus Wiseman, Apr 09 2025

nonn

The first x with a(x) > 1 but A382771(x) > 0 is a(216) = 4, A382771(216) = 4.

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, sum A056239.

			The prime indices of 216 are {1,1,1,2,2,2} and we have permutations:
  (1,1,1,2,2,2)
  (1,2,1,2,1,2)
  (2,1,2,1,2,1)
  (2,2,2,1,1,1)
so a(216) = 4.
The prime indices of 25920 are {1,1,1,1,1,1,2,2,2,2,3} and we have permutations:
  (1,2,1,2,1,2,1,2,1,3,1)
  (1,2,1,2,1,2,1,3,1,2,1)
  (1,2,1,2,1,3,1,2,1,2,1)
  (1,2,1,3,1,2,1,2,1,2,1)
  (1,3,1,2,1,2,1,2,1,2,1)
so a(25920) = 5.

The restriction to signature representatives (A181821) is A382858, distinct A382773.
The restriction to factorials is A335407, distinct A382774.
For distinct instead of equal run-lengths we have A382771.
For run-sums instead of run-lengths we have A382877, distinct A382876.
Positions of first appearances are A382878.
Positions of 0 are A382879.
Positions of terms > 1 are A383089.
Positions of 1 are A383112.
A003963 gives product of prime indices.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294.
A304442 counts partitions with equal run-sums, ranks A353833.
A164707 lists numbers whose binary expansion has all equal run-lengths, distinct A328592.
A353744 ranks compositions with equal run-lengths, counted by A329738.
Cf. A000720, A000961, A001221, A001222, A003242, A008480, A047966, A238130, A238279, A351201, A351293, A351295.

Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]], SameQ@@Length/@Split[#]&]],{n,0,100}]

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