A325535 -id:A325535 - cp's OEIS frontend

A335127 A multiset whose multiplicities are the prime indices of n is separable.

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 77, 80, 81, 84, 90, 96, 98, 99, 100, 105, 108, 110, 112, 120, 121, 125, 126, 128, 132, 135, 140, 143, 144, 147, 150, 154, 160, 162, 165

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is 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}.

			The sequence together with the corresponding multisets begins:
   1: {}
   2: {1}
   4: {1,2}
   6: {1,1,2}
   8: {1,2,3}
   9: {1,1,2,2}
  12: {1,1,2,3}
  15: {1,1,1,2,2}
  16: {1,2,3,4}
  18: {1,1,2,2,3}
  20: {1,1,1,2,3}
  24: {1,1,2,3,4}
  25: {1,1,1,2,2,2}
  27: {1,1,2,2,3,3}
  30: {1,1,1,2,2,3}

The complement is A335126.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are A325534.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Inseparable factorizations are A333487.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Cf. A056239, A106351, A112798, A114938, A292884, A335489, A335516, A335838.

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

A114938 Number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal.

Original entry on oeis.org

1, 0, 2, 30, 864, 39480, 2631600, 241133760, 29083420800, 4467125013120, 851371260364800, 197158144895712000, 54528028997584665600, 17752366094818747392000, 6720318485119046923315200, 2927066537906697348594432000, 1453437879238150456164433920000

Offset: 0

Hugo Pfoertner, Jan 08 2006

a(n) is also the number of (0,1)-matrices A=(a_ij) of size n X 2n such that each row has exactly two 1's and each column has exactly one 1 and with the restriction that no 1 stands on the line from a_11 to a_22. - Shanzhen Gao, Feb 24 2010
a(n) is the number of permutations of the multiset {1,1,2,2,...,n,n} with no fixed points. - Alexander Burstein, May 16 2020
Also the number of 2-uniform ordered set partitions of {1...2n} containing no two successive vertices in the same block. - Gus Wiseman, Jul 04 2020

			a(2) = 2 because there are two permutations of {1,1,2,2} avoiding equal consecutive terms: 1212 and 2121.

R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997. Chapter 2, Sieve Methods, Example 2.2.3, page 68.

Seiichi Manyama, Table of n, a(n) for n = 0..238 (terms 1..100 from Andrew Woods)
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.

Cf. A114939 = preferred seating arrangements of n couples.
Cf. A007060 = arrangements of n couples with no adjacent spouses; A007060(n) = 2^n * A114938(n) (this sequence).
Cf. A104548, A193638.
Cf. A278990 = number of loopless linear chord diagrams with n chords.
Cf. A000806 = Bessel polynomial y_n(-1).
The version for multisets with prescribed multiplicities is A335125.
The version for prime indices is A335452.
Anti-run compositions are counted by A003242.
Anti-run compositions are ranked by A333489.
Inseparable partitions are counted by A325535.
Inseparable partitions are ranked by A335448.
Separable partitions are counted by A325534.
Separable partitions are ranked by A335433.
Cf. A007716, A128695, A292884, A335126, A335434, A335451.
Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.
Row n=2 of A322093.

[1] cat [n le 2 select 2*(n-1) else n*(2*n-1)*Self(n-1) + (n-1)*n*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Aug 10 2015

Table[Sum[Binomial[n,i](2n-i)!/2^(n-i) (-1)^i,{i,0,n}],{n,0,20}]  (* Geoffrey Critzer, Jan 02 2013, and adapted to the extension by Stefano Spezia, Nov 15 2018 *)
Table[Length[Select[Permutations[Join[Range[n],Range[n]]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,5}] (* Gus Wiseman, Jul 04 2020 *)
A114938[n_] := ((2 n)! Hypergeometric1F1[-n, -2 n, -2]) / 2^n;
Array[A114938, 17, 0]  (* Peter Luschny, Sep 04 2025 *)

A114938(n)=sum(k=0, n, binomial(n, k)*(-1)^(n-k)*(n+k)!/2^k);
vector(20, n, A114938(n-1)) \\ Michel Marcus, Aug 10 2015

def A114938(n): return (-1)^n*sum(binomial(n,k)*factorial(n+k)//(-2)^k for k in range(n+1))
[A114938(n) for n in range(31)] # G. C. Greubel, Sep 26 2023

a(n) = Sum_{k=0..n} ((binomial(n, k)*(-1)^(n-k)*(n+k)!)/2^k).
a(n) = (-1)^n * n! * A000806(n), n>0. - Vladeta Jovovic, Nov 19 2009
a(n) = n*(2*n-1)*a(n-1) + (n-1)*n*a(n-2). - Vaclav Kotesovec, Aug 07 2013
a(n) ~ 2^(n+1)*n^(2*n)*sqrt(Pi*n)/exp(2*n+1). - Vaclav Kotesovec, Aug 07 2013
a(n) = n! * A278990(n). - Alexander Burstein, May 16 2020
From G. C. Greubel, Sep 26 2023: (Start)
a(n) = (-1)^n * (i/e)*sqrt(2/Pi) * n! * BesselK(n+1/2, -1).
a(n) = [n! * (1/x) * p_{n+1}(x)]|A104548%20for%20p">{x= -1} (See A104548 for p{n}(x)).
E.g.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * erfi((1+x)/sqrt(2*x)).
Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(sqrt(1-2*x))/sqrt(1-2*x).
Sum_{n >= 0} a(n)*x^n/(n!*(n+1)!) = ( 1 - exp(-1 + sqrt(1-2*x)) )/x. (End)
a(n) = ((2*n)!/2^n) * hypergeom([-n], [-2*n], -2]) = A007060(n) / 2^n. - Peter Luschny, Sep 04 2025

a(0)=1 prepended by Seiichi Manyama, Nov 15 2018

A345166 Number of separable integer partitions of n without an alternating permutation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 5, 6, 7, 10, 14, 18, 21, 27, 35, 42, 54, 65, 78, 95, 117, 140, 170, 202, 239, 286, 343, 401, 476, 562, 660, 775, 910, 1056, 1241, 1444, 1678, 1948, 2267, 2615, 3031, 3502, 4036, 4647, 5356, 6143, 7068, 8101, 9274, 10613, 12151, 13856

Offset: 0

Gus Wiseman, Jun 13 2021

nonn

A partition is separable if it has an anti-run permutation (no adjacent parts equal).
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 permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
The partitions counted by this sequence are those with 2m-1 parts with m being the multiplicity of a part which is neither the smallest or largest part. For example, 4322221 is such a partition since the multiplicity of 2 is 4, the total number of parts is 7, and 2 is neither the smallest or largest part. - Andrew Howroyd, Jan 15 2024

			The a(10) = 1 through a(16) = 6 partitions:
    32221  42221  52221  62221    43331    43332    53332
                         3222211  72221    53331    63331
                                  4222211  82221    92221
                                           3322221  4322221
                                           5222211  6222211
                                                    322222111

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Allowing alternating permutations gives A325534, ranked by A335433.
Not requiring separability gives A345165, ranked by A345171.
Permutations of this type are ranked by A345169.
The Heinz numbers of these partitions are A345173.
Numbers with a factorization of this type are A348609.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325535 counts inseparable partitions, ranked by A335448.
A344654 counts non-twin partitions w/o alt permutation, rank A344653.
A345162 counts normal partitions w/o alt permutation, complement A345163.
A345170 counts partitions w/ alt permutation, ranked by A345172.
Cf. A000070, A103919, A335126, A344604, A344607, A344615, A344740, A344742, A345164, A345166, A345168, A345192, A348379.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]!={}&&Select[Permutations[#],wigQ]=={}&]],{n,0,15}]

The Heinz numbers of these partitions are A345173 = A345171 /\ A335433.

a(n) = A325534(n) - A345170(n). - Andrew Howroyd, Jan 15 2024

a(26) onwards from Andrew Howroyd, Jan 15 2024

A347706 Number of factorizations of n that are not a twin (x*x) nor have an alternating permutation.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 28 2021

nonn

First differs from A348381 at a(216) = 4, A348381(216) = 3.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
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 permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of sets.

			The a(n) factorizations for n = 96, 192, 2160, 576:
  2*2*2*12      3*4*4*4         3*3*3*80       4*4*4*9
  2*2*2*2*6     2*2*2*24        6*6*6*10       2*2*2*72
  2*2*2*2*2*3   2*2*2*2*12      2*2*2*270      2*2*2*2*36
                2*2*2*2*2*6     2*3*3*3*40     2*2*2*2*4*9
                2*2*2*2*3*4     2*2*2*2*135    2*2*2*2*6*6
                2*2*2*2*2*2*3   2*2*2*2*3*45   2*2*2*2*2*18
                                2*2*2*2*5*27   2*2*2*2*3*12
                                2*2*2*2*9*15   2*2*2*2*2*2*9
                                               2*2*2*2*2*3*6
                                               2*2*2*2*2*2*3*3

Wikipedia, Alternating permutation

Positions of nonzero terms are A046099.
Partitions of this type are counted by A344654, ranked by A344653.
Partitions not of this type are counted by A344740, ranked by A344742.
The complement is counted by A347050, without twins A348379.
The version for compositions is A348377.
The version allowing twins is A348380.
The inseparable case is A348381.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations of sets.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347438 counts factorizations with alternating product 1, additive A119620.
A348610 counts alternating ordered factorizations.
Cf. A049774, A325535, A336103, A344614, A347437, A347439, A347442, A347456, A347463, A348382, A348383.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Function[f,Select[Permutations[f],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]=={}]]],{n,100}]

a(2^n) = A344654(n).

A336103 Number of separable multisets of size n covering an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 3, 5, 13, 24, 56, 108, 236, 464, 976, 1936, 3984, 7936, 16128, 32192, 64960, 129792, 260864, 521472, 1045760, 2091008, 4188160, 8375296, 16763904, 33525760, 67080192, 134156288, 268374016, 536739840, 1073610752, 2147205120, 4294688768, 8589344768, 17179279360, 34358493184

Offset: 0

Gus Wiseman, Jul 09 2020

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of its remaining multiplicities plus one. Hence a(n) is the number of compositions of n whose greatest part is at most one more than the sum of its other parts. For example, the a(1) = 1 through a(5) = 13 compositions are:
  (1)  (11)  (12)   (22)    (23)
             (21)   (112)   (32)
             (111)  (121)   (113)
                    (211)   (122)
                    (1111)  (131)
                            (212)
                            (221)
                            (311)
                            (1112)
                            (1121)
                            (1211)
                            (2111)
                            (11111)

			The a(1) = 1 through a(5) = 13 separable multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,2}
              {1,2,2}  {1,1,2,3}  {1,1,1,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,2,2}
                       {1,2,3,3}  {1,1,2,2,3}
                       {1,2,3,4}  {1,1,2,3,3}
                                  {1,1,2,3,4}
                                  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Michael De Vlieger, Table of n, a(n) for n = 0..3322
Index entries for linear recurrences with constant coefficients, signature (2,4,-8,-4,8).

The inseparable version is A336102.
The strong (weakly decreasing multiplicities) case is A336106.
Sequences covering an initial interval are A000670.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Separable partitions are A325534.
Inseparable partitions are A325535.
Inseparable factorizations are A333487.
Anti-run compositions are ranked by A333489.
Heinz numbers of inseparable partitions are A335448.
Cf. A001792, A019472, A025065, A049610, A052841, A106351, A269134, A292884, A335126, A335433, A335452.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sepQ[m_]:=Select[Permutations[m],!MatchQ[#,{_,x_,x_,_}]&]!={};
Table[Length[Select[allnorm[n],sepQ]],{n,0,5}]
(* or *)
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],With[{mx=Max@@#},mx<=1+Total[DeleteCases[#,mx,{1},1]]]&]],{n,0,15}] (* or *)
CoefficientList[Series[(x - 1) (2 x^5 + 7 x^4 - 5 x^2 + 1)/((2 x - 1) (2 x^2 - 1)^2), {x, 0, 36}], x] (* Michael De Vlieger, Apr 07 2021 *)

a(n) = 2^(n-1) - (floor(n/2)+1) * 2^(floor(n/2)-2) for n >= 2. - David A. Corneth, Jul 09 2020
From Chai Wah Wu, Apr 07 2021: (Start)
a(n) = 2*a(n-1) + 4*a(n-2) - 8*a(n-3) - 4*a(n-4) + 8*a(n-5) for n > 6.
G.f.: (x - 1)*(2*x^5 + 7*x^4 - 5*x^2 + 1)/((2*x - 1)*(2*x^2 - 1)^2). (End)

a(26)-a(36) from David A. Corneth, Jul 09 2020

A344742 Numbers whose prime factors have a permutation with no consecutive monotone triple, i.e., no triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Jun 12 2021

nonn

Differs from A335433 in having all squares of primes (A001248) and lacking 270 etc.
Also Heinz numbers of integer partitions that are either a twin (x,x) or have a wiggly permutation.
(1) 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.
(2) A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly permutations, even though it has anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The sequence of terms together with their prime indices begins:
      1: {}          18: {1,2,2}     36: {1,1,2,2}
      2: {1}         19: {8}         37: {12}
      3: {2}         20: {1,1,3}     38: {1,8}
      4: {1,1}       21: {2,4}       39: {2,6}
      5: {3}         22: {1,5}       41: {13}
      6: {1,2}       23: {9}         42: {1,2,4}
      7: {4}         25: {3,3}       43: {14}
      9: {2,2}       26: {1,6}       44: {1,1,5}
     10: {1,3}       28: {1,1,4}     45: {2,2,3}
     11: {5}         29: {10}        46: {1,9}
     12: {1,1,2}     30: {1,2,3}     47: {15}
     13: {6}         31: {11}        49: {4,4}
     14: {1,4}       33: {2,5}       50: {1,3,3}
     15: {2,3}       34: {1,7}       51: {2,7}
     17: {7}         35: {3,4}       52: {1,1,6}
For example, the prime factors of 120 are (2,2,2,3,5), with the two wiggly permutations (2,3,2,5,2) and (2,5,2,3,2), so 120 is in the sequence.

Positions of nonzero terms in A344606.
The complement is A344653, counted by A344654.
These partitions are counted by A344740.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001248 lists squares of primes.
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A011782 counts compositions.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A056239 adds up prime indices,  row sums of A112798.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts wiggly compositions with twins.
A345164 counts wiggly permutations of prime indices.
A345165 counts partitions without a wiggly permutation, ranked by A345171.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions.
Cf. A000070, A001222, A071321, A071322, A316523, A316524, A344605, A344614, A344616, A344652, A345163, A345166, A345167, A345173.

Select[Range[100],Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]!={}&]

Union of A345172 (wiggly) and A001248 (squares of primes).

A345173 Numbers whose multiset of prime factors is separable but has no alternating permutation.

Original entry on oeis.org

270, 378, 594, 702, 918, 1026, 1242, 1566, 1620, 1674, 1750, 1998, 2214, 2268, 2322, 2538, 2625, 2750, 2862, 3186, 3250, 3294, 3564, 3618, 3834, 3942, 4050, 4125, 4212, 4250, 4266, 4482, 4750, 4806, 4875, 5238, 5454, 5508, 5562, 5670, 5750, 5778, 5886, 6102

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

A multiset is separable if it has an anti-run permutation (no adjacent parts equal).
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 permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
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 terms together with their prime indices begin:
   270: {1,2,2,2,3}
   378: {1,2,2,2,4}
   594: {1,2,2,2,5}
   702: {1,2,2,2,6}
   918: {1,2,2,2,7}
  1026: {1,2,2,2,8}
  1242: {1,2,2,2,9}
  1566: {1,2,2,2,10}
  1620: {1,1,2,2,2,2,3}
  1674: {1,2,2,2,11}
  1750: {1,3,3,3,4}
  1998: {1,2,2,2,12}
  2214: {1,2,2,2,13}
  2268: {1,1,2,2,2,2,4}
  2322: {1,2,2,2,14}

The partitions with these Heinz numbers are counted by A345166.
Permutations of this type are ranked by A345169.
Numbers with a factorization of this type are counted by A348609.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A025047 counts alternating compositions, ascend A025048, descend A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices with twins.
A344740 counts twins and partitions with an alternating permutation.
A345164 counts alternating permutations of prime factors.
A345165 counts partitions without an alternating permutation.
A345170 counts partitions with an alternating permutation.
A345192 counts non-alternating compositions, without twins A348377.
A348379 counts factorizations with an alternating permutation.
Cf. A001222, A071321, A316524, A335126, A344614, A344616, A344652, A344653, A345163, A345168, A345193, A347706, A348380, A348613.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
sepQ[y_]:=!MatchQ[y,{_,x_,x_,_}];
Select[Range[1000],Select[Permutations[primeMS[#]],wigQ]=={}&&!Select[Permutations[primeMS[#]],sepQ]=={}&]

Equals A345171 /\ A335433.

A346697 Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 01 2021

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 prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 + 5 = 9.
The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 2 + 3 = 6.

The version for standard compositions is A209281(n+1) (even: A346633).
Subtracting the even version gives A316524 (reverse: A344616).
The even version is A346698.
The reverse version is A346699.
The even reverse version is A346700.
A000120 and A080791 count binary digits 1 and 0, with difference A145037.
A000302 counts compositions with odd alternating sum, ranked by A053738.
A001414 adds up prime factors, row sums of A027746.
A029837 adds up parts of standard compositions (alternating: A124754).
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A120452, A341446, A344614, A344617, A344653, A344654, A345957, A345958, A345959.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[First/@Partition[Append[primeMS[n],0],2]],{n,100}]

a(n) = A056239(n) - A346698(n).
a(n) = A316524(n) + A346698(n).
a(n odd omega) = A346699(n).
a(n even omega) = A346700(n).
A344616(n) = A346699(n) - A346700(n).

A335125 Number of anti-run permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 01 2020

nonn

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is 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}.

An anti-run is a sequence with no adjacent equal parts.

			The a(n) permutations for n = 2, 4, 42, 8, 30, 18:
  (1)  (12)  (1212131)  (123)  (121213)  (12123)
       (21)  (1213121)  (132)  (121231)  (12132)
             (1312121)  (213)  (121312)  (12312)
                        (231)  (121321)  (12321)
                        (312)  (123121)  (13212)
                        (321)  (131212)  (21213)
                               (132121)  (21231)
                               (212131)  (21312)
                               (213121)  (21321)
                               (312121)  (23121)
                                         (31212)
                                         (32121)

Positions of zeros are A335126.
Positions of nonzeros are A335127.
The version for the prime indices themselves is A335452.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Separable factorizations are A335434.
Inseparable partitions are ranked by A335448.
Patterns contiguously matched by compositions are A335457.
Strict permutations of prime indices are A335489.
Cf. A019472, A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335407, A335463, A335516, A335838.

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]],!MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

A345957 Number of divisors of n with exactly half as many prime factors as n, counting multiplicity.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 0, 0, 2, 2, 1, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 0, 2, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 0, 4, 0, 2, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 2, 0, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 16 2021

nonn

These divisors do not necessarily include the central divisors (A207375), and may not themselves be central.

			The a(n) divisors for selected n:
  n = 1:  6:  36:  60:  210:  840:  900:  1260:  1296:  3600:
     --------------------------------------------------------
      1   2    4    4     6     8    12     12     16     16
          3    6    6    10    12    18     18     24     24
               9   10    14    20    20     20     36     36
                   15    15    28    30     28     54     40
                         21    30    45     30     81     60
                         35    42    50     42            90
                               70    75     45           100
                              105           63           150
                                            70           225
                                           105

The case of powers of 2 is A000035.
Positions of even terms are A000037.
Positions of odd terms are A000290.
Positions of 0's are A026424.
Positions of 1's are A056798.
The rounded version is A096825.
The case of all divisors (not just 2) is A347042.
The smallest of these divisors is A347045 (rounded: A347043).
The greatest of these divisors is A347046 (rounded: A347044).
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A334997 counts chains of divisors of n by length.
Cf. A001227, A001414, A028260, A033676, A033677, A073093, A074206, A217581, A344653, A346697-A346704.

Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeOmega[n]/2&]],{n,100}]

a(n) = my(nb=bigomega(n)); sumdiv(n, d, 2*bigomega(d) == nb); \\ Michel Marcus, Aug 16 2021

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    divs = divisors(n)
    return sum(2*len(factorint(d, multiple=True)) == npf for d in divs)
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Aug 17 2021
(Python 3.8+)
from itertools import combinations
from math import prod, comb
from sympy import factorint
def A345957(n):
    if n == 1:
        return 1
    fs = factorint(n)
    elist = list(fs.values())
    q, r = divmod(sum(elist),2)
    k = len(elist)
    if r:
        return 0
    c = 0
    for i in range(k+1):
        m = (-1)**i
        for d in combinations(range(k),i):
            t = k+q-sum(elist[j] for j in d)-i-1
            if t >= 0:
                c += m*comb(t,k-1)
    return c # Chai Wah Wu, Aug 20 2021

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A345957(n):
    if n == 1:
        return 1
    fs = factorint(n,multiple=True)
    q, r = divmod(len(fs),2)
    return 0 if r else len(list(multiset_combinations(fs,q))) # Chai Wah Wu, Aug 20 2021

cp's OEIS Frontend

A335127 A multiset whose multiplicities are the prime indices of n is separable.

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A114938 Number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal.

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A345166 Number of separable integer partitions of n without an alternating permutation.

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A347706 Number of factorizations of n that are not a twin (x*x) nor have an alternating permutation.

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A336103 Number of separable multisets of size n covering an initial interval of positive integers.

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A344742 Numbers whose prime factors have a permutation with no consecutive monotone triple, i.e., no triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

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A345173 Numbers whose multiset of prime factors is separable but has no alternating permutation.

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