A347548 -id:A347548 - cp's OEIS frontend

A349795 Number of non-strict integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

0, 0, 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 39, 46, 61, 69, 90, 103, 131, 147, 185, 207, 259, 286, 355, 391, 482, 528, 644, 706, 858, 933, 1129, 1228, 1477, 1597, 1916, 2072, 2473, 2668, 3168, 3415, 4047, 4347, 5133, 5514, 6488, 6952, 8162, 8738, 10226, 10936

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

Also the number of 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 looks at the somewhat degenerate case where no strict increases are allowed. Equivalently, these are partitions that are weakly alternating but not strongly alternating.

			The a(2) = 1 through a(8) = 14 partitions:
  (11)  (111)  (22)    (221)    (33)      (322)      (44)
               (211)   (311)    (222)     (331)      (332)
               (1111)  (2111)   (411)     (511)      (422)
                       (11111)  (2211)    (2221)     (611)
                                (3111)    (4111)     (2222)
                                (21111)   (22111)    (3221)
                                (111111)  (31111)    (3311)
                                          (211111)   (5111)
                                          (1111111)  (22211)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

This is the restriction of A349060 to non-strict partitions.
The complement in non-strict partitions is A349796.
Permutations of prime factors of this type are counted by A349798.
The ordered version (compositions) is A349800, ranked by A349799.
These partitions are ranked by A350137.
A000041 counts integer partitions, non-strict A047967.
A001250 counts alternating permutations, complement A348615.
A025047 counts alternating compositions, also A025048 and A025049.
A096441 counts weakly alternating 0-appended partitions.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349053 counts non-weakly alternating compositions, complement A349052.
A349061 counts non-weakly alternating partitions, ranked by A349794.
A349801 counts non-alternating partitions.
Cf. A000070, A003242, A027383, A065033, A117298, A129852, A274230, A344614, A344615, A345165, A345167, A347548, A349056, A349057.

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

a(n > 0) = A349060(n) - A065033(n) = A349060(n) - floor(n/2).

a(n) = A047967(n) - A349796(n).

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).

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).

A347549 Number of partitions of n into 4 or more distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 38, 48, 59, 73, 89, 108, 130, 156, 185, 220, 259, 304, 356, 415, 482, 559, 645, 743, 854, 979, 1119, 1278, 1455, 1654, 1878, 2127, 2405, 2717, 3063, 3449, 3879, 4356, 4885, 5474, 6125, 6846, 7645, 8527, 9501, 10579

Offset: 10

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000009, A014591, A111133, A347548.

nmax = 60; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 4, nmax}], {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: Sum_{k>=4} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

a(n) = A000009(n) - A014591(n). - Vaclav Kotesovec, Sep 14 2021

A347574 Number of partitions of n into 7 or more distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 70, 89, 116, 146, 186, 231, 289, 354, 437, 530, 645, 775, 934, 1112, 1327, 1569, 1856, 2179, 2559, 2985, 3483, 4040, 4684, 5406, 6235, 7160, 8218, 9396, 10735, 12225, 13910, 15780, 17888, 20223, 22842, 25742, 28983, 32562

Offset: 28

Ilya Gutkovskiy, Sep 07 2021

nonn

Chai Wah Wu, Table of n, a(n) for n = 28..10000

Cf. A111133, A347548, A347549, A347572, A347573, A347575, A347576, A347577.

nmax = 77; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 7, nmax}], {x, 0, nmax}], x] // Drop[#, 28] &

G.f.: Sum_{k>=7} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

A350352 Products of three or more distinct prime numbers.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 426, 429, 430

Offset: 1

Gus Wiseman, Jan 11 2022

nonn

First differs from A336568 in lacking 420.

			The terms and their prime indices begin:
     30: {1,2,3}     182: {1,4,6}      285: {2,3,8}
     42: {1,2,4}     186: {1,2,11}     286: {1,5,6}
     66: {1,2,5}     190: {1,3,8}      290: {1,3,10}
     70: {1,3,4}     195: {2,3,6}      310: {1,3,11}
     78: {1,2,6}     210: {1,2,3,4}    318: {1,2,16}
    102: {1,2,7}     222: {1,2,12}     322: {1,4,9}
    105: {2,3,4}     230: {1,3,9}      330: {1,2,3,5}
    110: {1,3,5}     231: {2,4,5}      345: {2,3,9}
    114: {1,2,8}     238: {1,4,7}      354: {1,2,17}
    130: {1,3,6}     246: {1,2,13}     357: {2,4,7}
    138: {1,2,9}     255: {2,3,7}      366: {1,2,18}
    154: {1,4,5}     258: {1,2,14}     370: {1,3,12}
    165: {2,3,5}     266: {1,4,8}      374: {1,5,7}
    170: {1,3,7}     273: {2,4,6}      385: {3,4,5}
    174: {1,2,10}    282: {1,2,15}     390: {1,2,3,6}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

This is the squarefree case of A033942.
Including squarefree semiprimes gives A120944.
The squarefree complement consists of 1 and A167171.
These are the Heinz numbers of the partitions counted by A347548.
A000040 lists prime numbers (exactly 1 prime factor).
A005117 lists squarefree numbers.
A006881 lists squarefree numbers with exactly 2 prime factors.
A007304 lists squarefree numbers with exactly 3 prime factors.
A046386 lists squarefree numbers with exactly 4 prime factors.
Cf. A000009, A000111, A001250, A002808, A027383, A349796.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]>=3&]

is(n,f=factor(n))=my(e=f[,2]); #e>2 && vecmax(e)==1 \\ Charles R Greathouse IV, Jul 08 2022

list(lim)=my(v=List()); forsquarefree(n=30,lim\1, if(#n[2][,2]>2, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 08 2022

from sympy import factorint
def ok(n): f = factorint(n, multiple=True); return len(f) == len(set(f)) > 2
print([k for k in range(431) if ok(k)]) # Michael S. Branicky, Jan 14 2022

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A350352(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(3,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 11 2024

A347572 Number of partitions of n into 5 or more distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 72, 91, 112, 139, 168, 206, 246, 297, 353, 420, 494, 584, 682, 798, 927, 1077, 1243, 1437, 1649, 1894, 2166, 2475, 2817, 3207, 3636, 4121, 4658, 5261, 5926, 6673, 7494, 8412, 9425, 10550, 11788, 13166, 14677, 16352

Offset: 15

Ilya Gutkovskiy, Sep 07 2021

nonn

Cf. A111133, A347548, A347549, A347573, A347574, A347575, A347576, A347577.

nmax = 65; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 5, nmax}], {x, 0, nmax}], x] // Drop[#, 15] &

G.f.: Sum_{k>=5} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

A347573 Number of partitions of n into 6 or more distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 49, 65, 82, 105, 132, 165, 203, 251, 305, 371, 447, 537, 640, 763, 901, 1063, 1248, 1461, 1702, 1981, 2294, 2652, 3056, 3514, 4028, 4611, 5261, 5994, 6814, 7732, 8754, 9900, 11170, 12587, 14160, 15906, 17839, 19985, 22352

Offset: 21

Ilya Gutkovskiy, Sep 07 2021

nonn

Cf. A111133, A347548, A347549, A347572, A347574, A347575, A347576, A347577.

nmax = 70; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 6, nmax}], {x, 0, nmax}], x] // Drop[#, 21] &

G.f.: Sum_{k>=6} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

A347575 Number of partitions of n into 8 or more distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 73, 94, 123, 157, 201, 252, 318, 394, 489, 600, 735, 892, 1083, 1302, 1564, 1867, 2224, 2634, 3116, 3665, 4305, 5035, 5877, 6834, 7935, 9179, 10601, 12208, 14033, 16087, 18415, 21024, 23968, 27264, 30965, 35097, 39728, 44881

Offset: 36

Ilya Gutkovskiy, Sep 07 2021

nonn

Cf. A111133, A347548, A347549, A347572, A347573, A347574, A347576, A347577.

nmax = 85; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 8, nmax}], {x, 0, nmax}], x] // Drop[#, 36] &

G.f.: Sum_{k>=8} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

A347576 Number of partitions of n into 9 or more distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 55, 75, 97, 128, 164, 212, 267, 340, 423, 530, 654, 808, 986, 1207, 1460, 1768, 2123, 2549, 3037, 3620, 4284, 5069, 5965, 7012, 8203, 9590, 11160, 12975, 15029, 17388, 20048, 23092, 26513, 30408, 34781, 39734, 45278

Offset: 45

Ilya Gutkovskiy, Sep 07 2021

nonn

Cf. A111133, A347548, A347549, A347572, A347573, A347574, A347575, A347577.

nmax = 92; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 9, nmax}], {x, 0, nmax}], x] // Drop[#, 45] &

G.f.: Sum_{k>=9} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

cp's OEIS Frontend

A349795 Number of non-strict integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

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

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A347549 Number of partitions of n into 4 or more distinct parts.

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A347574 Number of partitions of n into 7 or more distinct parts.

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A350352 Products of three or more distinct prime numbers.

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A347572 Number of partitions of n into 5 or more distinct parts.

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A347573 Number of partitions of n into 6 or more distinct parts.

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A347575 Number of partitions of n into 8 or more distinct parts.

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