A067659 -id:A067659 - cp's OEIS frontend

A069288 Number of odd divisors of n <= sqrt(n).

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

Offset: 1

Reinhard Zumkeller, Mar 14 2002

nonn

a(n) = #{d : d = A182469(n,k), d <= A000196(n), k=1..A001227(n)}. - Reinhard Zumkeller, Apr 05 2015

			From _Gus Wiseman_, Feb 11 2021: (Start)
The inferior odd divisors for selected n are the columns below:
n: 1    9   30   90  225  315  630  945 1575 2835 4410 3465 8190 6930
  --------------------------------------------------------------------
   1    3    5    9   15   15   21   27   35   45   63   55   65   77
        1    3    5    9    9   15   21   25   35   49   45   63   63
             1    3    5    7    9   15   21   27   45   35   45   55
                  1    3    5    7    9   15   21   35   33   39   45
                       1    3    5    7    9   15   21   21   35   35
                            1    3    5    7    9   15   15   21   33
                                 1    3    5    7    9   11   15   21
                                      1    3    5    7    9   13   15
                                           1    3    5    7    9   11
                                                1    3    5    7    9
                                                     1    3    5    7
                                                          1    3    5
                                                               1    3
                                                                    1
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000196, A001227, A069289, A182469.
Positions of first appearances are A334853.
A055396 selects the least prime index.
A061395 selects the greatest prime index.
- Odd -
A000009 counts partitions into odd parts (A066208).
A026424 lists numbers with odd Omega.
A027193 counts odd-length partitions.
A067659 counts strict partitions of odd length (A030059).
- Inferior divisors -
A033676 selects the greatest inferior divisor.
A033677 selects the least superior divisor.
A038548 counts inferior divisors.
A060775 selects the greatest strictly inferior divisor.
A063538 lists numbers with a superior prime divisor.
A063539 lists numbers without a superior prime divisor.
A063962 counts inferior prime divisors.
A064052 lists numbers with a properly superior prime divisor.
A140271 selects the least properly superior divisor.
A217581 selects the greatest inferior divisor.
A333806 counts strictly inferior prime divisors.
Cf. A001055, A244991, A300272, A340101, A340607, A340832, A340854/A340855.

a069288 n = length $ takeWhile (<= a000196 n) $ a182469_row n
-- Reinhard Zumkeller, Apr 05 2015

odn[n_]:=Count[Divisors[n],?(OddQ[#]&&#<=Sqrt[n ]&)]; Array[odn,100] (* _Harvey P. Dale, Nov 04 2017 *)

a(n) = my(ir = sqrtint(n)); sumdiv(n, d, (d % 2) * (d <= ir)); \\ Michel Marcus, Jan 14 2014

G.f.: Sum_{n>=1} 1/(1-q^(2*n-1)) * q^((2*n-1)^2). [Joerg Arndt, Mar 04 2010]

A359894 Number of integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 10, 13, 20, 28, 49, 53, 93, 113, 145, 203, 287, 329, 479, 556, 724, 955, 1242, 1432, 1889, 2370, 2863, 3502, 4549, 5237, 6825, 8108, 9839, 12188, 14374, 16958, 21617, 25852, 30582, 36100, 44561, 51462, 63238, 73386, 85990, 105272, 124729

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(4) = 1 through a(8) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)
         (311)   (3111)   (331)     (422)
         (2111)  (21111)  (421)     (431)
                          (511)     (521)
                          (2221)    (611)
                          (3211)    (4211)
                          (4111)    (5111)
                          (22111)   (22211)
                          (31111)   (32111)
                          (211111)  (41111)
                                    (221111)
                                    (311111)
                                    (2111111)

The complement is counted by A240219.
These partitions are ranked by A359890, complement A359889.
The odd-length case is ranked by A359892, complement A359891.
The odd-length case is A359896, complement A359895.
The strict case is A359898, complement A359897.
The odd-length strict case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284 and A058398 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A326622 counts factorizations with integer mean, strict A328966.
A359893 and A359901 count partitions by median, odd-length A359902.
A359909 counts factorizations with the same mean as median, odd-len A359910.
Cf. A066571, A316313, A327472, A327482, A349156, A359906.

Table[Length[Select[IntegerPartitions[n],Mean[#]!=Median[#]&]],{n,0,30}]

A344608 Number of integer partitions of n with reverse-alternating sum < 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 7, 7, 14, 15, 27, 29, 49, 54, 86, 96, 146, 165, 242, 275, 392, 449, 623, 716, 973, 1123, 1498, 1732, 2274, 2635, 3411, 3955, 5059, 5871, 7427, 8620, 10801, 12536, 15572, 18065, 22267, 25821, 31602, 36617, 44533, 51560, 62338, 72105, 86716

Offset: 0

Gus Wiseman, May 30 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
Also the number of reversed of integer partitions of n with alternating sum < 0.
No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.
Is this sequence weakly increasing? Note: a(2n + 2) = A236914(n), a(2n) = A344743(n).
A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)    (42)    (43)      (53)      (54)
              (41)    (51)    (52)      (62)      (63)
              (2111)  (3111)  (61)      (71)      (72)
                              (2221)    (3221)    (81)
                              (3211)    (4211)    (3222)
                              (4111)    (5111)    (3321)
                              (211111)  (311111)  (4221)
                                                  (4311)
                                                  (5211)
                                                  (6111)
                                                  (222111)
                                                  (321111)
                                                  (411111)
                                                  (21111111)

The opposite version (rev-alt sum > 0) is A027193, ranked by A026424.
The strict case (for n > 2) is A067659 (odd bisection: A344650).
The Heinz numbers of these partitions are A119899 (complement: A344609).
The bisections are A236914 (odd) and A344743 (even).
The ordered version appears to be A294175 (even bisection: A008549).
The complement is counted by A344607 (even bisection: A344611).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A027187 counts partitions with alternating sum <= 0, ranked by A028260.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions with rev-alternating sum 2 (negative: A344741).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A000070, A000097, A000346, A003242, A006330, A071322, A239829, A239830, A344649, A344651, A344654/A344740, A344739.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]<0&]],{n,0,30}]

A333805 Number of odd divisors of n that are < sqrt(n).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

nonn

If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence counts strictly inferior odd divisors. - Gus Wiseman, Feb 26 2021

			The strictly inferior odd divisors of 945 are 1, 3, 5, 7, 9, 15, 21, 27, so a(945) = 8. - _Gus Wiseman_, Feb 27 2021

Antti Karttunen, Table of n, a(n) for n = 1..65537

Dominated by A001227 (number of odd divisors).
Strictly inferior divisors (not just odd) are counted by A056924.
The non-strict version is A069288.
These divisors add up to A070039.
The case of prime divisors is A333806.
The strictly superior version is A341594.
The case of squarefree divisors is A341596.
The superior version is A341675.
The case of prime-power divisors is A341677.
A006530 selects the greatest prime factor.
A020639 selects the smallest prime factor.
- Odd -
A000009 counts partitions into odd parts, ranked by A066208.
A026424 lists numbers with odd Omega.
A027193 counts odd-length partitions.
A067659 counts strict partitions of odd length, ranked by A030059.
- Inferior divisors -
A033676 selects the greatest inferior divisor.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A060775 selects the greatest strictly inferior divisor.
A341674 lists strictly inferior divisors.
Cf. A001248, A051283, A063538/A063539, A063962, A116883/A116882, A300272, A333807, A333809, A340832.

Table[DivisorSum[n, 1 &, # < Sqrt[n] && OddQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A333805(n) = sumdiv(n,d,(d%2)&&((d*d)Antti Karttunen, Nov 02 2022

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

Data section extended up to a(105) by Antti Karttunen, Nov 02 2022

A344609 Numbers whose alternating sum of prime indices is >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 30, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 102, 103, 105, 107

Offset: 1

Gus Wiseman, May 30 2021

nonn

Also Heinz numbers of partitions whose reverse-alternating sum is >= 0. These are partitions whose conjugate parts are all even or whose length is odd.
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 sequence of terms together with their prime indices begins:
      1: {}            20: {1,1,3}         45: {2,2,3}
      2: {1}           23: {9}             47: {15}
      3: {2}           25: {3,3}           48: {1,1,1,1,2}
      4: {1,1}         27: {2,2,2}         49: {4,4}
      5: {3}           28: {1,1,4}         50: {1,3,3}
      7: {4}           29: {10}            52: {1,1,6}
      8: {1,1,1}       30: {1,2,3}         53: {16}
      9: {2,2}         31: {11}            59: {17}
     11: {5}           32: {1,1,1,1,1}     61: {18}
     12: {1,1,2}       36: {1,1,2,2}       63: {2,2,4}
     13: {6}           37: {12}            64: {1,1,1,1,1,1}
     16: {1,1,1,1}     41: {13}            66: {1,2,5}
     17: {7}           42: {1,2,4}         67: {19}
     18: {1,2,2}       43: {14}            68: {1,1,7}
     19: {8}           44: {1,1,5}         70: {1,3,4}
For example, the prime indices of 70 are {1,3,4} with alternating sum 1 - 3 + 4 = 2, so 70 is in the sequence. On the other hand, the prime indices of 24 are {1,1,1,2} with alternating sum 1 - 1 + 1 - 2 = -1, so 24 is not in the sequence.

The opposite (nonpositive) version is A028260, counted by A027187.
The strict case (n > 0) is counted by A067659, odd bisection A344650.
Permutations of prime indices of these terms are counted by A116406.
Complement of A119899, Heinz numbers of the partitions counted by A344608.
Positions of nonnegative terms in A316524 or A344617.
Heinz numbers of the partitions counted by A344607.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions with alternating sum 1.
A000097 counts partitions with alternating sum 2.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum.
A120452 counts partitions with reverse-alternating sum 2.
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A335433/A335448 rank separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344612 counts partitions by sum and reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A001222, A001250, A003242, A005649, A026424, A071321/A071322, A124754, A239829, A343938, A344611, A344651, A344653/A344742, A344739.

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[100],ats[primeMS[#]]>=0&]

A340604 Heinz numbers of integer partitions of odd positive rank.

Original entry on oeis.org

3, 7, 10, 13, 15, 19, 22, 25, 28, 29, 33, 34, 37, 42, 43, 46, 51, 52, 53, 55, 61, 62, 63, 69, 70, 71, 76, 77, 78, 79, 82, 85, 88, 89, 93, 94, 98, 101, 105, 107, 113, 114, 115, 116, 117, 118, 119, 121, 123, 130, 131, 132, 134, 136, 139, 141, 146, 147, 148, 151

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.

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 sequence of partitions with their Heinz numbers begins:
      3: (2)         46: (9,1)       82: (13,1)
      7: (4)         51: (7,2)       85: (7,3)
     10: (3,1)       52: (6,1,1)     88: (5,1,1,1)
     13: (6)         53: (16)        89: (24)
     15: (3,2)       55: (5,3)       93: (11,2)
     19: (8)         61: (18)        94: (15,1)
     22: (5,1)       62: (11,1)      98: (4,4,1)
     25: (3,3)       63: (4,2,2)    101: (26)
     28: (4,1,1)     69: (9,2)      105: (4,3,2)
     29: (10)        70: (4,3,1)    107: (28)
     33: (5,2)       71: (20)       113: (30)
     34: (7,1)       76: (8,1,1)    114: (8,2,1)
     37: (12)        77: (5,4)      115: (9,3)
     42: (4,2,1)     78: (6,2,1)    116: (10,1,1)
     43: (14)        79: (22)       117: (6,2,2)

FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
These partitions are counted by A101707.
Allowing negative ranks gives A340692, counted by A340603.
The even version is A340605, counted by A101708.
The not necessarily odd case is A340787, counted by A064173.
A001222 gives number of prime indices.
A061395 gives maximum prime index.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A064173 counts partitions of negative rank (A340788).
A064174 counts partitions of nonnegative rank (A324562).
A064174 (also) counts partitions of nonpositive rank (A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts (A066208).
A027193 counts partitions of odd length (A026424).
A027193 (also) counts partitions of odd maximum (A244991).
A058695 counts partitions of odd numbers (A300063).
A067659 counts strict partitions of odd length (A030059).
A160786 counts odd-length partitions of odd numbers (A300272).
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340601, A340602, A340608, A340609, A340610.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[100],OddQ[rk[#]]&&rk[#]>0&]

A061395(a(n)) - A001222(a(n)) is odd and positive.

A340604 \/ A340605 = A340787.

A298118 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are odd.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 21, 40, 80, 159, 322, 657, 1356, 2816, 5896, 12407, 26267, 55861, 119331, 255878, 550665, 1188786, 2574006, 5588177, 12162141, 26529873, 57993624, 127020653, 278716336, 612617523, 1348680531, 2973564157, 6565313455, 14514675376

Offset: 1

Gus Wiseman, Jan 12 2018

nonn

			The a(6) = 6 trees: (((((o))))), (((ooo))), ((oo(o))), (oo((o))), (o(o)(o)), (ooooo).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000081, A000598, A003238, A004111, A027193, A032305, A067659, A290689, A291443, A297791, A298120.

orut[n_]:=orut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[orut/@c]]]/@Select[IntegerPartitions[n-1],OddQ[Length[#]]&]];
Table[Length[orut[n]],{n,15}]

a(n) ~ c * d^n / n^(3/2), where d = 2.30984417428419893876754252289588812511559... and c = 0.5598122522173731208680575003383895445787... - Vaclav Kotesovec, Jun 04 2019

a(24)-a(35) from Alois P. Heinz, Jan 12 2018

A090864 Complement of generalized pentagonal numbers (A001318).

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Jon Perry, Feb 12 2004

Also n for which A006906(n) is even, or equivalently n for which A000009(n) is even (since A006906 and A000009 have the same parity).

The number of partitions of a(n) into distinct parts with an even number of parts equals the number of such partitions with an odd number of parts: A067661(a(n)) = A067659(a(n)). See, e.g., the Freitag-Busam reference, p. 410 given in A036499. - Wolfdieter Lang, Jan 19 2016

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000009, A001318, A067659, A067661, A080995, A006906, A010815, A118303.

Cf. A085141, A111651.

Complement[Range[200], Select[Accumulate[Range[0,200]]/3, IntegerQ]] (* G. C. Greubel, Jun 06 2017 *)

a(n) = my(q,r); [q,r]=divrem(sqrtint(24*n),3); n + q + (r >= bitnegimply(1,q)); \\ Kevin Ryde, Sep 15 2024

from math import isqrt
def A090864(n):
    def f(x): return n+(m:=isqrt(24*x+1)+1)//6+(m-2)//6
    kmin, kmax = 0,1
    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 # Chai Wah Wu, Aug 29 2024

A080995(a(n)) = 0; A000009(a(n)) = A118303(n). - Reinhard Zumkeller, Apr 22 2006
A010815(a(n)) = A067661(a(n)) - A067659(a(n)) = 0, n >= 1. See a comment above. - Wolfdieter Lang, Jan 19 2016
a(n) = n+1 + A085141(n-1) + A111651(n). - Kevin Ryde, Sep 15 2024

More terms from Reinhard Zumkeller, Apr 22 2006
Edited by Ray Chandler, Dec 14 2011
Edited by Jon E. Schoenfield, Nov 25 2016

A344650 Number of strict odd-length integer partitions of 2n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 11, 16, 23, 32, 44, 61, 82, 111, 148, 195, 256, 334, 432, 557, 713, 908, 1152, 1455, 1829, 2291, 2859, 3554, 4404, 5440, 6697, 8222, 10066, 12288, 14964, 18176, 22023, 26625, 32117, 38656, 46432, 55661, 66592, 79523, 94793, 112792, 133984

Offset: 0

Gus Wiseman, Jun 05 2021

nonn

Also the number of strict integer partitions of 2n with reverse-alternating sum >= 0.

Also the number of reversed strict integer partitions of 2n with alternating sum >= 0.

			The a(1) = 1 through a(8) = 16 partitions:
  (2)  (4)  (6)      (8)      (10)     (12)     (14)      (16)
            (3,2,1)  (4,3,1)  (5,3,2)  (5,4,3)  (6,5,3)   (7,5,4)
                     (5,2,1)  (5,4,1)  (6,4,2)  (7,4,3)   (7,6,3)
                              (6,3,1)  (6,5,1)  (7,5,2)   (8,5,3)
                              (7,2,1)  (7,3,2)  (7,6,1)   (8,6,2)
                                       (7,4,1)  (8,4,2)   (8,7,1)
                                       (8,3,1)  (8,5,1)   (9,4,3)
                                       (9,2,1)  (9,3,2)   (9,5,2)
                                                (9,4,1)   (9,6,1)
                                                (10,3,1)  (10,4,2)
                                                (11,2,1)  (10,5,1)
                                                          (11,3,2)
                                                          (11,4,1)
                                                          (12,3,1)
                                                          (13,2,1)
                                                          (6,4,3,2,1)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

The Heinz numbers are the intersection of A030059 and A300061.
Allowing even length gives A035294 (non-strict: A058696).
Even bisection of A067659.
The opposite type of strict partition (even length and odd sum) is A343942.
The non-strict version is A236559 or A344611.
Row sums of A344649.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A124754 gives alternating sums of standard compositions (reverse: A344618).
A152146 interleaved with A152157 counts strict partitions by sum and alternating sum.
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A343941 counts strict partitions of 2n with reverse-alternating sum 4.
A344604 counts wiggly compositions with twins.
A344739 counts strict partitions by sum and reverse-alternating sum.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A000070, A000097, A027187, A114121, A239829, A239830, A344607, A344609, A344610, A344651, A344654.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(2*n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,30,2}]

Sum of odd-indexed terms in row 2n of A008289.

a(n) = A067659(2n).

A361849 Number of integer partitions of n such that the maximum is twice the median.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 3, 4, 7, 9, 9, 15, 16, 20, 26, 34, 37, 50, 55, 68, 86, 103, 117, 145, 168, 201, 236, 282, 324, 391, 449, 525, 612, 712, 818, 962, 1106, 1278, 1470, 1698, 1939, 2238, 2550, 2924, 3343, 3824, 4341, 4963, 5627, 6399, 7256, 8231, 9300

Offset: 1

Gus Wiseman, Apr 02 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(4) = 1 through a(11) = 9 partitions:
  211  2111  21111  421     422      4221      631        632
                    3211    221111   4311      4222       5321
                    22111   2111111  2211111   42211      5411
                    211111           21111111  322111     42221
                                               2221111    43211
                                               22111111   332111
                                               211111111  22211111
                                                          221111111
                                                          2111111111
For example, the partition (3,2,1,1) has maximum 3 and median 3/2, so is counted under a(7).

For minimum instead of median we have A118096.
For length instead of median we have A237753.
This is the equal case of A361848.
For mean instead of median we have A361853.
These partitions have ranks A361856.
For "greater" instead of "equal" we have A361857, allowing equality A361859.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
A361860 counts partitions with minimum equal to median.
Cf. A013580, A027193, A053263, A061395, A067659, A111907, A116608, A237755, A240219, A361851.

Table[Length[Select[IntegerPartitions[n],Max@@#==2*Median[#]&]],{n,30}]

cp's OEIS Frontend

A069288 Number of odd divisors of n <= sqrt(n).

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A359894 Number of integer partitions of n whose parts do not have the same mean as median.

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A344608 Number of integer partitions of n with reverse-alternating sum < 0.

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A333805 Number of odd divisors of n that are < sqrt(n).

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A344609 Numbers whose alternating sum of prime indices is >= 0.

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A340604 Heinz numbers of integer partitions of odd positive rank.

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A298118 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are odd.

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A090864 Complement of generalized pentagonal numbers (A001318).

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