A349795 -id:A349795 - cp's OEIS frontend

A349057 Numbers k such that the k-th composition in standard order is not weakly alternating.

37, 46, 52, 53, 69, 75, 78, 92, 93, 101, 104, 105, 107, 110, 116, 117, 133, 137, 139, 142, 150, 151, 156, 157, 165, 174, 180, 181, 184, 185, 186, 187, 190, 197, 200, 201, 203, 206, 208, 209, 210, 211, 214, 215, 220, 221, 229, 232, 233, 235, 238, 244, 245, 261

Offset: 1

Gus Wiseman, Dec 04 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms and corresponding compositions begin:
   37: (3,2,1)
   46: (2,1,1,2)
   52: (1,2,3)
   53: (1,2,2,1)
   69: (4,2,1)
   75: (3,2,1,1)
   78: (3,1,1,2)
   92: (2,1,1,3)
   93: (2,1,1,2,1)
  101: (1,3,2,1)
  104: (1,2,4)
  105: (1,2,3,1)
  107: (1,2,2,1,1)
  110: (1,2,1,1,2)
  116: (1,1,2,3)
  117: (1,1,2,2,1)

The strong case is A345168, complement A345167, counted by A345192.
The strong anti-run case is A345169, counted by A345195.
Including all non-anti-runs gives A348612, complement A333489.
These compositions are counted by A349053, complement A349052.
The directed cases are counted by A129852 (incr.) and A129853 (decr.).
The complement for patterns is A349058, strong A345194.
The complement for ordered factorizations is A349059, strong A348610.
Partitions of this type are counted by A349061, complement A349060.
Partitions of this type are ranked by A349794.
Non-strict partitions of this type are counted by A349796.
Permutations of prime indices of this type are counted by A349797.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions, complement A261983.
A011782 counts compositions.
A025047 counts alternating/wiggly compositions, directed A025048, A025049.
A345164 counts alternating permutations of prime indices, weak A349056.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349054 counts strict alternating compositions.
Cf. A001700, A096441, A128761, A344615, A344654, A345173, A348613, A349051, A349794, A349795, A349799.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Select[Range[0,100],!whkQ[stc[#]]&&!whkQ[-stc[#]]&]

A349060 Number of integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 18, 22, 29, 35, 45, 53, 68, 77, 98, 112, 140, 157, 195, 218, 270, 298, 367, 404, 495, 542, 658, 721, 873, 949, 1145, 1245, 1494, 1615, 1934, 2091, 2492, 2688, 3188, 3436, 4068, 4369, 5155, 5537, 6511, 6976, 8186, 8763, 10251, 10962

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

Also the number of weakly alternating integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed.

			The a(1) = 1 through a(7) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

John Tyler Rascoe, Table of n, a(n) for n = 0..300

Alternating: A025047, ranked by A345167, also A025048 and A025049.
The strong case is A065033, ranked by A167171.
A directed version is A096441.
Non-alternating: A345192, ranked by A345168.
Weakly alternating: A349052, also A129852 and A129853.
Non-weakly alternating: A349053, ranked by A349057.
A version for ordered factorizations is A349059, strong A348610.
The complement is counted by A349061, strong A349801.
These partitions are ranked by the complement of A349794.
The non-strict case is A349795.
A000041 counts integer partitions, ordered A011782.
A001250 counts alternating permutations, complement A348615.
A344604 counts alternating compositions with twins.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
Cf. A000070, A002865, A003242, A027383, A114901, A117298, A117989, A274230, A345163, A349056, A349796.

Table[Length[Select[IntegerPartitions[n], SameQ@@#||And@@EvenQ/@Take[Length/@Split[#],{2,-2}]&]],{n,0,30}]

A_x(N)={my(x='x+O('x^N), g= 1 + sum(i=1, N, (x^i/(1-x^i)) * (1 + sum(j=i+1, N-i, (x^j/((1-x^j))) / prod(k=1, j-i-1, 1-x^(2*(i+k)))))));
Vec(g)}
A_x(52) \\ John Tyler Rascoe, Mar 20 2024

G.f.: 1 + Sum_{i>0} (x^i/(1-x^i)) * (1 + Sum_{j>i} (x^j/(1-x^j)) / Product_{k=1..j-i-1} (1-x^(2*(i+k)))). - John Tyler Rascoe, Mar 20 2024

A349061 Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 13, 21, 32, 48, 67, 99, 133, 185, 245, 333, 432, 574, 732, 957, 1208, 1554, 1941, 2468, 3060, 3844, 4731, 5893, 7204, 8898, 10816, 13268, 16043, 19546, 23523, 28497, 34150, 41147, 49106, 58892, 70020, 83597, 99047, 117778, 139087

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

Also the number of non-weakly alternating integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed.

			The a(6) = 1 through a(10) = 13 partitions:
  (321)  (421)   (431)    (432)     (532)
         (3211)  (521)    (531)     (541)
                 (4211)   (621)     (631)
                 (32111)  (3321)    (721)
                          (4311)    (4321)
                          (5211)    (5311)
                          (42111)   (6211)
                          (321111)  (32221)
                                    (33211)
                                    (43111)
                                    (52111)
                                    (421111)
                                    (3211111)

The strong version for compositions is A345192, ranked by A345168.
The version for compositions is A349053, ranked by A349057.
The complement is counted by A349060.
These partitions are ranked by A349794.
The non-strict case is A349796, complement A349795.
The strong case is A349801.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A025047 counts alternating compositions, ranked by A345167.
A025048 and A025049 count directed alternating compositions.
A096441 counts weakly alternating 0-appended partitions.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349052 counts weakly alternating compositions.
A349056 counts weakly alternating permutations of prime indices.
A349798 counts weakly but not strongly alternating perms of prime indices.
Cf. A002865, A027383, A117298, A117989, A129852, A129853, A274230, A333213, A344615, A345165, A349059.

Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&!And@@EvenQ/@Take[Length/@Split[#],{2,-2}]&]],{n,0,30}]

A349054 Number of alternating strict compositions of n. Number of alternating (up/down or down/up) permutations of strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 35, 41, 59, 75, 103, 155, 193, 255, 339, 443, 569, 841, 1019, 1365, 1743, 2295, 2879, 3785, 5151, 6417, 8301, 10625, 13567, 17229, 21937, 27509, 37145, 45425, 58345, 73071, 93409, 115797, 147391, 182151, 229553, 297061, 365625

Offset: 0

Gus Wiseman, Dec 21 2021

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.
The case starting with an increase (or decrease, it doesn't matter in the enumeration) is counted by A129838.

			The a(1) = 1 through a(7) = 11 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)
                          (3,2)  (4,2)    (3,4)
                          (4,1)  (5,1)    (4,3)
                                 (1,3,2)  (5,2)
                                 (2,1,3)  (6,1)
                                 (2,3,1)  (1,4,2)
                                 (3,1,2)  (2,1,4)
                                          (2,4,1)
                                          (4,1,2)

Wikipedia, Alternating permutation

Ranking sequences are put in parentheses below.
This is the strict case of A025047/A025048/A025049 (A345167).
This is the alternating case of A032020 (A233564).
The unordered case (partitions) is A065033.
The directed case is A129838.
A001250 = alternating permutations (A349051), complement A348615 (A350250).
A003242 = Carlitz (anti-run) compositions, complement A261983.
A011782 = compositions, unordered A000041.
A345165 = partitions without an alternating permutation (A345171).
A345170 = partitions with an alternating permutation (A345172).
A345192 = non-alternating compositions (A345168).
A345195 = non-alternating anti-run compositions (A345169).
A349800 = weakly but not strongly alternating compositions (A349799).
A349052 = weakly alternating compositions, complement A349053 (A349057).
Cf. A000111, A008965, A015723, A114901, A128761, A129852, A129853, A218074, A333213, A344614, A345164, A345194, A349060, A349795.

g:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(g(o-1+j, u-j), j=1..u))
    end:
b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 2, 0), b(n-k, k)+b(n-k, k-1)))
    end:
a:= n-> add(b(n, k)*g(k, 0), k=0..floor((sqrt(8*n+1)-1)/2))-1:
seq(a(n), n=0..46);  # Alois P. Heinz, Dec 22 2021

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],wigQ]],{n,0,15}]

a(n) = 2 * A129838(n) - 1.

G.f.: Sum_{n>0} A001250(n)*x^(n*(n+1)/2)/Product_{k=1..n}(1-x^k).

A349800 Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.

Original entry on oeis.org

0, 0, 1, 1, 4, 9, 16, 33, 62, 113, 205, 373, 664, 1190, 2113, 3744, 6618, 11683, 20564, 36164, 63489, 111343, 195042, 341357, 596892, 1042976, 1821179, 3178145, 5543173, 9663545, 16839321, 29332231, 51075576, 88908912, 154722756, 269186074, 468221264

Offset: 0

Gus Wiseman, Dec 16 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

This sequence counts compositions that are weakly but not strongly alternating; also weakly alternating non-anti-run compositions.

			The a(2) = 1 through a(6) = 16 compositions:
  (1,1)  (1,1,1)  (2,2)      (1,1,3)      (3,3)
                  (1,1,2)    (1,2,2)      (1,1,4)
                  (2,1,1)    (2,2,1)      (2,2,2)
                  (1,1,1,1)  (3,1,1)      (4,1,1)
                             (1,1,1,2)    (1,1,1,3)
                             (1,1,2,1)    (1,1,2,2)
                             (1,2,1,1)    (1,1,3,1)
                             (2,1,1,1)    (1,3,1,1)
                             (1,1,1,1,1)  (2,2,1,1)
                                          (3,1,1,1)
                                          (1,1,1,1,2)
                                          (1,1,1,2,1)
                                          (1,1,2,1,1)
                                          (1,2,1,1,1)
                                          (2,1,1,1,1)
                                          (1,1,1,1,1,1)

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

This is the weakly alternating case of A345192, ranked by A345168.
The case of partitions is A349795, ranked by A350137.
The version counting permutations of prime indices is A349798.
These compositions are ranked by A349799.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions, ranked by A333489.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A261983 = non-anti-run compositions, ranked by A348612.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A345173 = non-alternating anti-run partitions, ranked by A345166.
A345195 = non-alternating anti-run compositions, ranked by A345169.
A348377 = non-alternating non-twin compositions.
A349801 = non-alternating partitions, ranked by A289553.
Weakly alternating:
- A349052 = compositions, directed A129852/A129853, complement A349053.
- A349056 = permutations of prime indices, complement A349797.
- A349057 = complement of standard composition numbers (too dense).
- A349058 = patterns, complement A350138.
- A349059 = ordered factorizations, complement A350139.
- A349060 = partitions, complement A349061.
Cf. A008965, A011782, A027383, A096441, A274230, A333213, A344614, A344615, A348382, A348613, A349796, A350140.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],(whkQ[#]||whkQ[-#])&&!wigQ[#]&]],{n,0,10}]

a(n) = A349052(n) - A025047(n). - Andrew Howroyd, Jan 31 2024

a(21) onwards from Andrew Howroyd, Jan 31 2024

A349796 Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 8, 15, 23, 37, 52, 80, 109, 156, 208, 289, 378, 509, 654, 865, 1098, 1425, 1789, 2290, 2852, 3603, 4450, 5569, 6830, 8467, 10321, 12701, 15393, 18805, 22678, 27535, 33057, 39908, 47701, 57304, 68226, 81572, 96766, 115212, 136201

Offset: 0

Gus Wiseman, Dec 25 2021

nonn

Also the number of non-weakly alternating non-strict integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence involves the somewhat degenerate case where no strict increases are allowed.

			The a(7) = 1 through a(11) = 15 partitions:
  (3211)  (4211)   (3321)    (5311)     (4322)
          (32111)  (4311)    (6211)     (4421)
                   (5211)    (32221)    (5411)
                   (42111)   (33211)    (6311)
                   (321111)  (43111)    (7211)
                             (52111)    (42221)
                             (421111)   (43211)
                             (3211111)  (53111)
                                        (62111)
                                        (322211)
                                        (332111)
                                        (431111)
                                        (521111)
                                        (4211111)
                                        (32111111)

Counting all non-strict partitions gives A047967.
Signatures of this type are counted by A274230, complement A027383.
The strict instead of non-strict version is A347548, ranked by A350352.
The version for compositions allowing strict is A349053, ranked by A349057.
Allowing strict partitions gives A349061, complement A349060.
The complement in non-strict partitions is A349795.
These partitions are ranked by A350140 = A349794 \ A005117.
A000041 = integer partitions, strict A000009.
A001250 = alternating permutations, complement A348615.
A003242 = Carlitz (anti-run) compositions.
A025047 = alternating compositions, ranked by A345167.
A025048/A025049 = directed alternating compositions.
A096441 = weakly alternating 0-appended partitions.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349798 = weakly but not strongly alternating permutations of prime indices.
Cf. A000111, A002865, A117298, A117989, A129852, A129853, A345165, A345192, A349054, A349059, A349801.

whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[IntegerPartitions[n],!whkQ[#]&&!whkQ[-#]&&!UnsameQ@@#&]],{n,0,30}]

a(n) = A349061(n) - A347548(n).

A349798 Number of weakly alternating ordered prime factorizations of n with at least two adjacent equal parts.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 2, 2, 0, 0, 0, 5, 1, 0, 0, 2, 0, 0, 0

Offset: 1

Gus Wiseman, Dec 14 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence counts permutations of prime factors that are weakly but not strongly alternating. Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

			Using prime indices instead of factors, the a(n) ordered prime factorizations for selected n are:
n = 4    12    24     48      90     120     192       240      270
   ------------------------------------------------------------------
    11   112   1112   11112   1223   11132   1111112   111132   12232
         211   1121   11121   1322   11213   1111121   111213   13222
               1211   11211   2213   11312   1111211   111312   21223
               2111   12111   2231   21113   1112111   112131   21322
                      21111   3122   21311   1121111   113121   22132
                              3221   23111   1211111   121113   22213
                                     31112   2111111   121311   22231
                                     31211             131112   22312
                                                       131211   23122
                                                       211131   23221
                                                       213111   31222
                                                       231111   32212
                                                       311121
                                                       312111

Wikipedia, Alternating permutation

This is the weakly but not strictly alternating case of A008480.
Including alternating (in fact, anti-run) permutations gives A349056.
These partitions are counted by A349795, ranked by A350137.
A complementary version is A349796, ranked by A350140.
The version for compositions is A349800, ranked by A349799.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A335452 = anti-run ordered prime factorizations.
A344652 = ordered prime factorizations w/o weakly increasing triples.
A345164 = alternating ordered prime factorizations, with twins A344606.
A345194 = alternating patterns, with twins A344605.
A349052/A129852/A129853 = weakly alternating compositions.
A349053 = non-weakly alternating compositions, ranked by A349057.
A349060 = weakly alternating partitions, complement A349061.
A349797 = non-weakly alternating ordered prime factorizations.
Cf. A005649, A096441, A335433, A335448, A344614, A344615, A345166, A345173, A345195, A348609, A349794, A349801.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Permutations[primeMS[n]],(whkQ[#]||whkQ[-#])&&MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

A349799 Numbers k such that the k-th composition in standard order is weakly alternating but has at least two adjacent equal parts.

Original entry on oeis.org

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 47, 51, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 79, 83, 84, 85, 86, 87, 90, 91, 94, 95, 99, 100, 103, 106, 111, 112, 113, 114, 115, 118, 119, 120, 121, 122, 123, 124, 125

Offset: 1

Gus Wiseman, Dec 15 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
This sequence ranks compositions that are weakly but not strongly alternating.

			The terms and corresponding compositions begin:
   3: (1,1)
   7: (1,1,1)
  10: (2,2)
  11: (2,1,1)
  14: (1,1,2)
  15: (1,1,1,1)
  19: (3,1,1)
  21: (2,2,1)
  23: (2,1,1,1)
  26: (1,2,2)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)

Wikipedia, Alternating permutation

Partitions of this type are counted by A349795, ranked by A350137.
Permutations of prime indices of this type are counted by A349798.
These compositions are counted by A349800.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions, ranked by A333489.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A261983 = non-anti-run compositions, ranked by A348612.
A345164 = alternating permutations of prime indices, with twins A344606.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A345166 = separable partitions with no alternations, ranked by A345173.
A345192 = non-alternating compositions, ranked by A345168.
A345195 = non-alternating anti-run compositions, ranked by A345169.
A349052/A129852/A129853 = weakly alternating compositions.
A349053 = non-weakly alternating compositions, ranked by A349057.
A349056 = weak alternations of prime indices, complement A349797.
A349060 = weak alternations of partitions, complement A349061.
Cf. A005649, A049774, A096441, A128761, A344615, A345163, A349054, A349058, A349796, A349801, A350140.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Select[Range[0,100],(whkQ[stc[#]]||whkQ[-stc[#]])&&MatchQ[stc[#],{_,x_,x_,_}]&]

Equals A345168 \ A349057 = A348612 \ A349057.

A349794 Numbers whose prime signature has an odd term other than the first or last.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 102, 105, 110, 114, 120, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 182, 186, 190, 195, 204, 210, 220, 222, 228, 230, 231, 238, 240, 246, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285, 286, 290, 294, 300, 308

Offset: 1

Gus Wiseman, Dec 06 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also numbers whose multiset of prime factors is not weakly alternating, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict decreases are allowed.

			The terms and their prime indices begin:
   30: {1,2,3}
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  114: {1,2,8}
  120: {1,1,1,2,3}
  130: {1,3,6}
  132: {1,1,2,5}
  138: {1,2,9}

The complement for compositions is A025047, ranked by A345167.
Signatures of this type are counted by A274230, complement A027383.
The strong case is A289553, complement A167171.
The strong case for compositions is A345192, ranked by A345168.
The version for compositions is A349053, ranked by A349057.
These partitions are counted by A349061, complement A349060, strong A349801.
The non-strict case is counted by A349795.
A001250 counts alternating permutations, complement A348615.
A096441 counts weakly alternating partitions if 0 is appended.
A345164 counts alternating permutations of prime indices, weak A349056.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349052 counts weakly alternating compositions.
A349059 counts weakly alternating ordered factorizations, strong A348610.
Cf. A003242, A112798, A114901, A117298, A128761, A129852, A129853, A344614, A344615, A345165, A349796, A349798.

Select[Range[100],PrimeNu[#]>1&&!And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]

A349801 Number of integer partitions of n into three or more parts or into two equal parts.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 11, 18, 25, 37, 50, 71, 94, 128, 168, 223, 288, 376, 480, 617, 781, 991, 1243, 1563, 1945, 2423, 2996, 3704, 4550, 5589, 6826, 8333, 10126, 12293, 14865, 17959, 21618, 25996, 31165, 37318, 44562, 53153, 63239, 75153, 89111, 105535, 124730

Offset: 0

Gus Wiseman, Dec 23 2021

nonn

This sequence arose as the following degenerate case. If we define a sequence to be alternating if it is alternately strictly increasing and strictly decreasing, starting with either, then a(n) is the number of non-alternating integer partitions of n. Under this interpretation:
- The non-strict case is A047967, weak A349796, weak complement A349795.
- The complement is counted by A065033(n) = ceiling(n/2) for n > 0.
- These partitions are ranked by A289553 \ {1}, complement A167171 \/ {1}.
- The version for compositions is A345192, ranked by A345168.
- The weak version for compositions is A349053, ranked by A349057.
- The weak version is A349061, complement A349060, ranked by A349794.

			The a(2) = 1 through a(7) = 11 partitions:
  (11)  (111)  (22)    (221)    (33)      (322)
               (211)   (311)    (222)     (331)
               (1111)  (2111)   (321)     (421)
                       (11111)  (411)     (511)
                                (2211)    (2221)
                                (3111)    (3211)
                                (21111)   (4111)
                                (111111)  (22111)
                                          (31111)
                                          (211111)
                                          (1111111)

A000041 counts partitions, ordered A011782.
A001250 counts alternating permutations, complement A348615.
A004250 counts partitions into three or more parts, strict A347548.
A025047/A025048/A025049 count alternating compositions, ranked by A345167.
A096441 counts weakly alternating 0-appended partitions.
A345165 counts partitions w/ no alternating permutation, complement A345170.
Cf. A000070, A001700, A002865, A117298, A117989, A102726, A128761, A345162, A345163, A345166, A349798.

Table[Length[Select[IntegerPartitions[n],MatchQ[#,{x_,x_}|{,,__}]&]],{n,0,10}]

a(1) = 0; a(n > 0) = A000041(n) - ceiling(n/2).

cp's OEIS Frontend

A349057 Numbers k such that the k-th composition in standard order is not weakly alternating.

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A349060 Number of integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

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A349061 Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.

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A349054 Number of alternating strict compositions of n. Number of alternating (up/down or down/up) permutations of strict integer partitions of n.

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A349800 Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.

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A349796 Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.

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A349798 Number of weakly alternating ordered prime factorizations of n with at least two adjacent equal parts.

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A349799 Numbers k such that the k-th composition in standard order is weakly alternating but has at least two adjacent equal parts.

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