A086543 -id:A086543 - cp's OEIS frontend

A365067 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n whose odd parts sum to k, for k ranging from mod(n,2) to n in steps of 2.

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 4, 3, 4, 3, 5, 5, 3, 4, 4, 6, 5, 6, 6, 5, 8, 7, 5, 6, 8, 6, 10, 7, 10, 9, 10, 8, 12, 11, 7, 10, 12, 12, 10, 15, 11, 14, 15, 15, 16, 12, 18, 15, 11, 14, 20, 18, 20, 15, 22, 15, 22, 21, 25, 24, 24, 18, 27

Offset: 0

Gus Wiseman, Oct 16 2023

The version for all k = 0..n is A113685 (including zeros).

			Triangle begins:
   1
   1
   1  1
   1  2
   2  1  2
   2  2  3
   3  2  2  4
   3  4  3  5
   5  3  4  4  6
   5  6  6  5  8
   7  5  6  8  6 10
   7 10  9 10  8 12
  11  7 10 12 12 10 15
  11 14 15 15 16 12 18
  15 11 14 20 18 20 15 22
  15 22 21 25 24 24 18 27
Row n = 8 counts the following partitions:
  (8)     (611)    (431)     (521)      (71)
  (62)    (4211)   (41111)   (332)      (53)
  (44)    (22211)  (3221)    (32111)    (5111)
  (422)            (221111)  (2111111)  (3311)
  (2222)                                (311111)
                                        (11111111)
Row n = 9 counts the following partitions:
  (81)     (63)      (54)       (72)        (9)
  (621)    (6111)    (522)      (5211)      (711)
  (441)    (432)     (4311)     (3321)      (531)
  (4221)   (42111)   (411111)   (321111)    (51111)
  (22221)  (3222)    (32211)    (21111111)  (333)
           (222111)  (2211111)              (33111)
                                            (3111111)
                                            (111111111)

Row sums are A000041.
The version including all k is A113685, even version A113686.
Column k = 1 is A119620.
The even version and the reverse version are both A174713.
For odd-indexed instead of odd parts we have A346697, even version A346698.
The corresponding rank statistic is A366528, even version A366531.
A000009 counts partitions into odd parts, ranks A066208.
A086543 counts partitions with odd parts, ranks A366322.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A035363, A045931, A053253, A066967, A130780, A171966, A241638, A268335, A325698, A366533.

Table[Length[Select[IntegerPartitions[n], Total[Select[#,OddQ]]==k&]],{n,0,15},{k,Mod[n,2],n,2}]

T(n,k) = A000009(k) * A000041((n-k)/2).

A347448 Number of integer partitions of n with alternating product > 1.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 12, 17, 25, 35, 49, 66, 90, 120, 161, 209, 275, 355, 460, 585, 750, 946, 1199, 1498, 1881, 2335, 2909, 3583, 4430, 5428, 6666, 8118, 9912, 12013, 14586, 17592, 21252, 25525, 30695, 36711, 43956, 52382, 62469, 74173, 88132, 104303, 123499

Offset: 0

Gus Wiseman, Sep 16 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(2) = 1 through a(7) = 12 partitions:
  (2)  (3)   (4)    (5)     (6)      (7)
       (21)  (31)   (32)    (42)     (43)
             (211)  (41)    (51)     (52)
                    (311)   (222)    (61)
                    (2111)  (321)    (322)
                            (411)    (421)
                            (3111)   (511)
                            (21111)  (2221)
                                     (3211)
                                     (4111)
                                     (31111)
                                     (211111)

The strict case is A000009, except that a(0) = a(1) = 0.
Allowing any alternating product >= 1 gives A000041, reverse A344607.
Ranked by A028983 (reverse A347465), which has complement A028982.
The complement is counted by A119620, reverse A347443.
The multiplicative version is A339890, weak A347456, reverse A347705.
The even-length case is A344608.
Allowing any integer reverse-alternating product gives A347445.
Allowing any integer alternating product gives A347446.
The reverse version is A347449, also the odd-length case.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347461 counts possible alternating products of partitions.
Cf. A000070, A086543, A100824, A236913, A325534, A325535, A339846, A344654, A345196, A347440, A347444, A347462.

a:= n-> (p-> p(n)-p(iquo(n, 2)))(combinat[numbpart]):
seq(a(n), n=0..63);  # Alois P. Heinz, Oct 04 2021

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],altprod[#]>1&]],{n,0,30}]

a(n) = A000041(n) - A119620(n).

A366531 Sum of even prime indices of n.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 4, 0, 4, 0, 0, 2, 6, 4, 2, 0, 0, 4, 8, 0, 6, 0, 0, 2, 0, 6, 6, 4, 10, 2, 0, 0, 2, 0, 4, 4, 12, 8, 8, 0, 0, 6, 14, 0, 4, 0, 0, 2, 8, 0, 2, 6, 16, 6, 0, 4, 10, 10, 0, 2, 18, 0, 8, 0, 6, 2, 0, 0, 2, 4, 20, 4, 0, 12, 2, 8, 4, 8, 22, 0, 8, 0, 0, 6

Offset: 1

Gus Wiseman, Oct 22 2023

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 198 are {1,2,2,5}, so a(198) = 2+2 = 4.

Zeros are A066208, counted by A000009.
The triangle for the odd version is A113685, without zeros A365067.
The triangle for this statistic is A113686, without zeros A174713.
The odd version is A366528.
The halved version is A366533.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A239261 counts partitions with sum of odd parts = sum of even parts.
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

Table[Total[Cases[FactorInteger[n], {p_?(EvenQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

a(n) = A056239(n) - A366528(n).

A366533 Sum of even prime indices of n divided by 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 1, 3, 2, 1, 0, 0, 2, 4, 0, 3, 0, 0, 1, 0, 3, 3, 2, 5, 1, 0, 0, 1, 0, 2, 2, 6, 4, 4, 0, 0, 3, 7, 0, 2, 0, 0, 1, 4, 0, 1, 3, 8, 3, 0, 2, 5, 5, 0, 1, 9, 0, 4, 0, 3, 1, 0, 0, 1, 2, 10, 2, 0, 6, 1, 4, 2, 4, 11, 0, 4, 0, 0, 3, 0, 7

Offset: 1

Gus Wiseman, Oct 23 2023

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 198 are {1,2,2,5}, so a(198) = (2+2)/2 = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Zeros are A066208, counted by A000009.
The triangle for this statistic (without zeros) is A174713.
The un-halved odd version is A366528.
The un-halved version is A366531.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A365067 counts partitions by sum of odd parts (without zeros).
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

f:= proc(n) local F,t;
  F:= map(t -> [numtheory:-Pi(t[1]),t[2]], ifactors(n)[2]);
  add(`if`(t[1]::even, t[1]*t[2]/2, 0), t=F)
end proc:
map(f, [$1..100]); # Robert Israel, Nov 22 2023

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Select[prix[n],EvenQ]]/2,{n,100}]

a(n) = A366531(n)/2.

A347449 Number of integer partitions of n with reverse-alternating product > 1.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 10, 11, 20, 22, 37, 41, 66, 75, 113, 129, 190, 218, 310, 358, 497, 576, 782, 908, 1212, 1411, 1851, 2156, 2793, 3255, 4163, 4853, 6142, 7159, 8972, 10451, 12989, 15123, 18646, 21689, 26561, 30867, 37556, 43599, 52743, 61161, 73593

Offset: 0

Gus Wiseman, Sep 16 2021

nonn

All such partitions have odd length.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			The a(2) = 1 through a(9) = 11 partitions:
  (2)  (3)  (4)    (5)    (6)      (7)      (8)        (9)
            (211)  (311)  (222)    (322)    (332)      (333)
                          (321)    (421)    (422)      (432)
                          (411)    (511)    (431)      (522)
                          (21111)  (31111)  (521)      (531)
                                            (611)      (621)
                                            (22211)    (711)
                                            (32111)    (32211)
                                            (41111)    (42111)
                                            (2111111)  (51111)
                                                       (3111111)

The strict case is A067659, except that a(0) = a(1) = 0.
The even bisection is A236559.
The non-reverse multiplicative version is A339890, weak A347456.
The case of >= 1 instead of > 1 is A344607.
The opposite version is A344608, also the non-reverse even-length case.
The complement is counted by A347443, non-reverse A119620.
Allowing any integer reverse-alternating product gives A347445.
Allowing any integer alternating product gives A347446.
Reverse version of A347448; also the odd-length case.
The Heinz numbers of these partitions are the complement of A347450.
The multiplicative version (factorizations) is A347705.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A100824 counts partitions of n with alternating sum <= 1.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347462 counts possible reverse-alternating products of partitions.
Cf. A000070, A008549, A086543, A182616, A236913, A325534, A325535, A344611, A347442, A347444, A347447, A347453, A347461, A347465.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],altprod[Reverse[#]]>1&]],{n,0,30}]

a(n) = A344607(n) - A119620(n).

A366322 Heinz numbers of integer partitions containing at least one odd part. Numbers divisible by at least one prime of odd index.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, Oct 14 2023

nonn

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:
    2: {1}
    4: {1,1}
    5: {3}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}

The complement is A066207, counted by A035363.
For all odd parts we have A066208, counted by A000009.
Partitions of this type are counted by A086543.
For even instead of odd we have A324929, counted by A047967.
A031368 lists primes of odd index.
A112798 list prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
Cf. A000720, A001222, A003963, A257992, A318400, A324927, A358137.

Select[Range[100],Or@@OddQ/@PrimePi/@First/@FactorInteger[#]&]

A257991(a(n)) > 0.

A352142 Numbers whose prime factorization has all odd indices and all odd exponents.

Original entry on oeis.org

1, 2, 5, 8, 10, 11, 17, 22, 23, 31, 32, 34, 40, 41, 46, 47, 55, 59, 62, 67, 73, 82, 83, 85, 88, 94, 97, 103, 109, 110, 115, 118, 125, 127, 128, 134, 136, 137, 146, 149, 155, 157, 160, 166, 167, 170, 179, 184, 187, 191, 194, 197, 205, 206, 211, 218, 227, 230

Offset: 1

Gus Wiseman, Mar 18 2022

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions with all odd parts and all odd multiplicities, counted by A117958.

			The terms together with their prime indices begin:
   1 = 1
   2 = prime(1)
   5 = prime(3)
   8 = prime(1)^3
  10 = prime(1) prime(3)
  11 = prime(5)
  17 = prime(7)
  22 = prime(1) prime(5)
  23 = prime(9)
  31 = prime(11)
  32 = prime(1)^5
  34 = prime(1) prime(7)
  40 = prime(1)^3 prime(3)

David A. Corneth, Table of n, a(n) for n = 1..10000

The restriction to primes is A031368.
The first condition alone is A066208, counted by A000009.
These partitions are counted by A117958.
The squarefree case is A258116, even A258117.
The second condition alone is A268335, counted by A055922.
The even-even version is A352141 counted by A035444.
A000290 = exponents all even, counted by A035363.
A056166 = exponents all prime, counted by A055923.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even prime exponents, odd A162642.
A257991 counts odd prime indices, even A257992.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A351979 = odd indices with even exponents, counted by A035457.
A352140 = even indices with odd exponents, counted by A055922 aerated.
A352143 = odd indices with odd conjugate indices, counted by A053253 aerated.
Cf. A000720, A028260, A055396, A061395, A106529, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698, A325700.

Select[Range[100],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]

from itertools import count, islice
from sympy import primepi, factorint
def A352142_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:all(map(lambda x:x[1]%2 and primepi(x[0])%2, factorint(k).items())),count(max(startvalue,1)))
A352142_list = list(islice(A352142_gen(),30)) # Chai Wah Wu, Mar 18 2022

Intersection of A066208 and A268335.
A257991(a(n)) = A001222(a(n)).
A162642(a(n)) = A001221(a(n)).
A257992(a(n)) = A162641(a(n)) = 0.

A349159 Numbers whose sum of prime indices is twice their alternating sum.

Original entry on oeis.org

1, 12, 63, 66, 112, 190, 255, 325, 408, 434, 468, 609, 805, 832, 931, 946, 1160, 1242, 1353, 1380, 1534, 1539, 1900, 2035, 2067, 2208, 2296, 2387, 2414, 2736, 3055, 3108, 3154, 3330, 3417, 3509, 3913, 4185, 4340, 4503, 4646, 4650, 4664, 4864, 5185, 5684, 5863

Offset: 1

Gus Wiseman, Nov 23 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 alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their alternating sum.

			The terms and their prime indices begin:
     1: ()
    12: (2,1,1)
    63: (4,2,2)
    66: (5,2,1)
   112: (4,1,1,1,1)
   190: (8,3,1)
   255: (7,3,2)
   325: (6,3,3)
   408: (7,2,1,1,1)
   434: (11,4,1)
   468: (6,2,2,1,1)
   609: (10,4,2)
   805: (9,4,3)
   832: (6,1,1,1,1,1,1)
   931: (8,4,4)
   946: (14,5,1)
  1160: (10,3,1,1,1)

These partitions are counted by A000712 up to 0's.
An ordered version is A348614, negative A349154.
The negative version is A348617.
The reverse version is A349160, counted by A006330 up to 0's.
A025047 counts alternating or wiggly compositions, complement A345192.
A027193 counts partitions with rev-alt sum > 0, ranked by A026424.
A034871, A097805, and A345197 count compositions by alternating sum.
A035363 = partitions with alt sum 0, ranked by A066207, complement A086543.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A103919 counts partitions by alternating sum, reverse A344612.
A116406 counts compositions with alternating sum >= 0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344607 counts partitions with rev-alt sum >= 0, ranked by A344609.
A346697 adds up odd-indexed prime indices.
A346698 adds up even-indexed prime indices.
Cf. A000070, A000290, A001700, A028260, A045931, A120452, A195017, A241638, A257991, A257992, A325698, A345958, A349155.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[1000],Total[primeMS[#]]==2*ats[primeMS[#]]&]

A056239(a(n)) = 2*A316524(a(n)).

A346697(a(n)) = 3*A346698(a(n)).

A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

Original entry on oeis.org

1, 4, 16, 25, 64, 100, 121, 256, 289, 400, 484, 529, 625, 961, 1024, 1156, 1600, 1681, 1936, 2116, 2209, 2500, 3025, 3481, 3844, 4096, 4489, 4624, 5329, 6400, 6724, 6889, 7225, 7744, 8464, 8836, 9409, 10000, 10609, 11881, 12100, 13225, 13924, 14641, 15376

Offset: 1

Gus Wiseman, Mar 11 2022

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).

			The terms together with their prime indices begin:
     1: 1
     4: prime(1)^2
    16: prime(1)^4
    25: prime(3)^2
    64: prime(1)^6
   100: prime(1)^2 prime(3)^2
   121: prime(5)^2
   256: prime(1)^8
   289: prime(7)^2
   400: prime(1)^4 prime(3)^2
   484: prime(1)^2 prime(5)^2
   529: prime(9)^2
   625: prime(3)^4
   961: prime(11)^2
  1024: prime(1)^10
  1156: prime(1)^2 prime(7)^2
  1600: prime(1)^6 prime(3)^2
  1681: prime(13)^2
  1936: prime(1)^4 prime(5)^2

Amiram Eldar, Table of n, a(n) for n = 1..10000

The second condition alone (exponents all even) is A000290, counted by A035363.
The distinct prime factors of terms all come from A031368.
These partitions are counted by A035457 or A000009 aerated.
The first condition alone (indices all odd) is A066208, counted by A000009.
The squarefree square roots are A258116, even A258117.
A056166 = exponents all prime, counted by A055923.
A066207 = indices all even, counted by complement of A086543.
A076610 = indices all prime, counted by A000607.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even exponents, odd A162642.
A257991 counts odd indices, even A257992.
A268335 = exponents all odd, counted by A055922.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A352140 = even indices with odd exponents, counted by A055922 (aerated).
A352141 = even indices with even exponents, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, A325698, A325700.

Select[Range[1000],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]

from sympy import factorint, primepi
def ok(n):
    return all(primepi(p)%2==1 and e%2==0 for p, e in factorint(n).items())
print([k for k in range(15500) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Squares of elements of A066208.
Intersection of A066208 and A000290.
A257991(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
A162642(a(n)) = A257992(a(n)) = 0.
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022

A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 6, 12, 14, 26, 31, 51, 61, 95, 114, 169, 202, 289, 347, 481, 576, 782, 936, 1244, 1487, 1946, 2323, 2997, 3570, 4551, 5414, 6827, 8103, 10127, 11997, 14866, 17575, 21619, 25507, 31166, 36692, 44563, 52362, 63240, 74152, 89112, 104281, 124731

Offset: 0

Gus Wiseman, Sep 19 2023

nonn

Also the number of integer partitions of n with no two possibly equal parts summing to n.

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (111)  (1111)  (221)    (222)     (322)      (332)
                           (311)    (411)     (331)      (521)
                           (2111)   (2211)    (421)      (611)
                           (11111)  (21111)   (511)      (2222)
                                    (111111)  (2221)     (3221)
                                              (3211)     (3311)
                                              (4111)     (5111)
                                              (22111)    (22211)
                                              (31111)    (32111)
                                              (211111)   (221111)
                                              (1111111)  (311111)
                                                         (2111111)
                                                         (11111111)

First condition alone is A058984, complement A004526, ranks A100959.
Second condition alone is A086543, complement A035363, ranks !A344415.
The complement is counted by A238628.
The strict case is A365826, complement A365659.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A140106 counts strict partitions of length 2, complement A365827.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A005408, A008967, A068911, A365377, A365544, A365543.

Table[Length[Select[IntegerPartitions[n],Length[#]!=2&&FreeQ[#,n/2]&]],{n,0,15}]

from sympy import npartitions
def A365825(n): return npartitions(n)-(m:=n>>1)-(0 if n&1 else npartitions(m)-1) # Chai Wah Wu, Sep 23 2023

Heinz numbers are A100959 /\ !A344415.

a(n) = A000041(n)-(n-1)/2 if n is odd. a(n) = A000041(n)-n/2-A000041(n/2)+1 if n is even. - Chai Wah Wu, Sep 23 2023

a(31)-a(47) from Chai Wah Wu, Sep 23 2023

cp's OEIS Frontend

A365067 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n whose odd parts sum to k, for k ranging from mod(n,2) to n in steps of 2.

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A347448 Number of integer partitions of n with alternating product > 1.

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A366531 Sum of even prime indices of n.

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A366533 Sum of even prime indices of n divided by 2.

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A347449 Number of integer partitions of n with reverse-alternating product > 1.

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A366322 Heinz numbers of integer partitions containing at least one odd part. Numbers divisible by at least one prime of odd index.

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A352142 Numbers whose prime factorization has all odd indices and all odd exponents.

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A349159 Numbers whose sum of prime indices is twice their alternating sum.

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