A058695 -id:A058695 - cp's OEIS frontend

A371840 Number of integer partitions of n with non-biquanimous multiplicities.

0, 1, 2, 2, 4, 5, 8, 11, 16, 21, 31, 40, 55, 72, 97, 124, 165, 209, 271, 343, 441, 547, 700, 866, 1089, 1345, 1679, 2050, 2546, 3099, 3814, 4622, 5654, 6811, 8297, 9957, 12039, 14409, 17355, 20666, 24793, 29432, 35133, 41598, 49474, 58360, 69197, 81395, 96124

Offset: 0

Gus Wiseman, Apr 18 2024

nonn

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The partition y = (6,2,1,1) has multiplicities (1,1,2), which are biquanimous because we have the partition ((1,1),(2)), so y is not counted under a(10).
The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (321)     (421)      (422)
                            (11111)  (411)     (511)      (431)
                                     (3111)    (2221)     (521)
                                     (21111)   (4111)     (611)
                                     (111111)  (22111)    (2222)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The complement for parts is counted by A002219 aerated, ranks A357976.
These partitions have Heinz numbers A371782.
For parts we have A371795, ranks A371731, bisections A006827, A058695.
The complement is counted by A371839, ranks A371781.
A237258 = biquanimous strict partitions, ranks A357854, complement A371794.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A035470, A064914, A305551, A321142, A365543, A365925, A367094.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[IntegerPartitions[n], !biqQ[Length/@Split[#]]&]],{n,0,30}]

A342082 Numbers with an inferior odd divisor > 1.

Original entry on oeis.org

9, 12, 15, 18, 21, 24, 25, 27, 30, 33, 35, 36, 39, 40, 42, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 80, 81, 84, 85, 87, 90, 91, 93, 95, 96, 98, 99, 100, 102, 105, 108, 110, 111, 112, 114, 115, 117, 119, 120, 121, 123, 125

Offset: 1

Gus Wiseman, Mar 06 2021

nonn

We define a divisor d|n to be inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.

Numbers n with an odd prime factor <= sqrt(n). - Chai Wah Wu, Mar 09 2021

			The divisors > 1 of 72 are {2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, of which {3, 9} are odd and {2, 3, 4, 6, 8} are inferior, with intersection {3}, so 72 is in the sequence.

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

The strictly inferior version is the same with A001248 removed.
Positions of terms > 1 in A069288.
The superior version is A116882, with complement A116883.
The complement is A342081.
A006530 selects the greatest prime factor.
A020639 selects the smallest prime factor.
A038548 counts superior (or inferior) divisors, with strict case A056924.
- Odd -
A000009 counts partitions into odd parts, ranked by A066208.
A001227 counts odd divisors.
A026424 lists numbers with odd Omega.
A027193 counts odd-length partitions.
A058695 counts partitions of odd numbers.
A067659 counts strict partitions of odd length, ranked by A030059.
A340101 counts factorizations into odd factors; A340102 also has odd length.
A340854/A340855 cannot/can be factored with odd minimum factor.
A341594 counts strictly superior odd divisors
A341675 counts superior odd divisors.
- Inferior: A033676, A066839, A161906.
- Superior: A033677, A051283, A059172, A063538/A063539.
- Strictly Inferior A333805, A341674.
- Strictly Superior: A064052/A048098, A341645/A341646.
Cf. A000005, A000203, A001055, A001221, A001222, A001414, A207375, A244991, A300272, A340832, A340931.

Select[Range[100],Function[n,Select[Divisors[n]//Rest,OddQ[#]&&#<=n/#&]!={}]]

is(n) = #select(x -> x > 2 && x^2 <= n, factor(n)[, 1]) > 0; \\ Amiram Eldar, Nov 01 2024

from sympy import primefactors
A342082_list = [n for n in range(1,10**3) if len([p for p in primefactors(n) if p > 2 and p*p <= n]) > 0] # Chai Wah Wu, Mar 09 2021

A111295 Number of partitions of 3n+1.

Original entry on oeis.org

1, 5, 15, 42, 101, 231, 490, 1002, 1958, 3718, 6842, 12310, 21637, 37338, 63261, 105558, 173525, 281589, 451276, 715220, 1121505, 1741630, 2679689, 4087968, 6185689, 9289091, 13848650, 20506255, 30167357, 44108109, 64112359, 92669720, 133230930, 190569292

Offset: 0

Parthasarathy Nambi, Nov 01 2005

nonn

Old name was: P(3*n + 1) where P(m) is the unrestricted partition of m and n = 1,2,3,... .

a(n) is also the number of partitions of 4n-2 that include n as a part; see Comment at A000041. - Clark Kimberling, Mar 03 2014

			If n=25 then P(3*25 + 1) = 9289091.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000041, A016777, A058695.

Mathematica
```
Table[PartitionsP[3 n + 1], {n, 0, 10}]
```

a(n) = numbpart(3*n+1); \\ Michel Marcus, Mar 04 2014

a(n) = A000041(A016777(n)). - Michel Marcus, Sep 01 2021

Better name from Clark Kimberling, Mar 02 2014

A340929 Heinz numbers of integer partitions of odd negative rank.

Original entry on oeis.org

4, 12, 16, 18, 27, 40, 48, 60, 64, 72, 90, 100, 108, 112, 135, 150, 160, 162, 168, 192, 225, 240, 243, 250, 252, 256, 280, 288, 352, 360, 375, 378, 392, 400, 420, 432, 448, 528, 540, 567, 588, 600, 625, 630, 640, 648, 672, 700, 768, 792, 810, 832, 880, 882

Offset: 1

Gus Wiseman, Jan 29 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
       4: (1,1)             150: (3,3,2,1)
      12: (2,1,1)           160: (3,1,1,1,1,1)
      16: (1,1,1,1)         162: (2,2,2,2,1)
      18: (2,2,1)           168: (4,2,1,1,1)
      27: (2,2,2)           192: (2,1,1,1,1,1,1)
      40: (3,1,1,1)         225: (3,3,2,2)
      48: (2,1,1,1,1)       240: (3,2,1,1,1,1)
      60: (3,2,1,1)         243: (2,2,2,2,2)
      64: (1,1,1,1,1,1)     250: (3,3,3,1)
      72: (2,2,1,1,1)       252: (4,2,2,1,1)
      90: (3,2,2,1)         256: (1,1,1,1,1,1,1,1)
     100: (3,3,1,1)         280: (4,3,1,1,1)
     108: (2,2,2,1,1)       288: (2,2,1,1,1,1,1)
     112: (4,1,1,1,1)       352: (5,1,1,1,1,1)
     135: (3,2,2,2)         360: (3,2,2,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A101707.
The positive version is A101707 (A340604).
The even version is A101708 (A340930).
The not necessarily odd version is A064173 (A340788).
A001222 counts prime factors.
A027193 counts partitions of odd length (A026424).
A047993 counts balanced partitions (A106529).
A058695 counts partitions of odd numbers (A300063).
A061395 selects the maximum prime index.
A063995/A105806 count partitions by Dyson rank.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324516 counts partitions with rank equal to maximum minus minimum part (A324515).
A324518 counts partitions with rank equal to greatest part (A324517).
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A003114, A056239, A096401, A117193, A117409, A325134, A326845, A340604, A340605, A340787, A340854/A340855.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[2,100],OddQ[rk[#]]&&rk[#]<0&]

For all terms, A061395(a(n)) - A001222(a(n)) is odd and negative.

A346634 Number of strict odd-length integer partitions of 2n + 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 14, 19, 27, 38, 52, 71, 96, 128, 170, 224, 293, 380, 491, 630, 805, 1024, 1295, 1632, 2048, 2560, 3189, 3958, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29250, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937

Offset: 0

Gus Wiseman, Aug 01 2021

nonn

			The a(0) = 1 through a(7) = 14 partitions:
  (1)  (3)  (5)  (7)      (9)      (11)     (13)      (15)
                 (4,2,1)  (4,3,2)  (5,4,2)  (6,4,3)   (6,5,4)
                          (5,3,1)  (6,3,2)  (6,5,2)   (7,5,3)
                          (6,2,1)  (6,4,1)  (7,4,2)   (7,6,2)
                                   (7,3,1)  (7,5,1)   (8,4,3)
                                   (8,2,1)  (8,3,2)   (8,5,2)
                                            (8,4,1)   (8,6,1)
                                            (9,3,1)   (9,4,2)
                                            (10,2,1)  (9,5,1)
                                                      (10,3,2)
                                                      (10,4,1)
                                                      (11,3,1)
                                                      (12,2,1)
                                                      (5,4,3,2,1)

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

Odd bisection of A067659, which is ranked by A030059.
The even version is the even bisection of A067661.
The case of all odd parts is counted by A069911 (non-strict: A078408).
The non-strict version is A160786, ranked by A340931.
The non-strict even version is A236913, ranked by A340784.
The even-length version is A343942 (non-strict: A236914).
The even-sum version is A344650 (non-strict: A236559 or A344611).
A000009 counts partitions with all odd parts, ranked by A066208.
A000009 counts strict partitions, ranked by A005117.
A027193 counts odd-length partitions, ranked by A026424.
A027193 counts odd-maximum partitions, ranked by A244991.
A058695 counts partitions of odd numbers, ranked by A300063.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Other cases of odd length:
- A024429 set partitions
- A089677 ordered set partitions
- A166444 compositions
- A174726 ordered factorizations
- A332304 strict compositions
- A339890 factorizations
Cf. A000700, A008289, A047993, A072233, A106529, A168659, A218171, A340102, A340604, A340607.

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+1$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021

Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,15}]

More terms from Alois P. Heinz, Aug 05 2021

A229724 Triangular array read by rows: T(n,k) is the number of partitions of n in which the greatest odd part is equal to 2k-1; n >= 1, 1 <= k <= ceiling(n/2).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 4, 2, 1, 4, 3, 1, 7, 5, 2, 1, 7, 6, 3, 1, 12, 10, 5, 2, 1, 12, 12, 7, 3, 1, 19, 18, 11, 5, 2, 1, 19, 22, 14, 7, 3, 1, 30, 31, 21, 11, 5, 2, 1, 30, 37, 27, 15, 7, 3, 1, 45, 52, 38, 22, 11, 5, 2, 1, 45, 61, 48, 29, 15, 7, 3, 1, 67, 82, 66, 41

Offset: 1

Geoffrey Critzer, Sep 28 2013

Row sums are A086543.

			1;
1;
2,   1;
2,   1;
4,   2,  1;
4,   3,  1;
7,   5,  2,  1;
7,   6,  3,  1;
12, 10,  5,  2, 1;
12, 12,  7,  3, 1;
19, 18, 11,  5, 2, 1;
19, 22, 14,  7, 3, 1;
30, 31, 21, 11, 5, 2, 1;
T(7,2) = 5 because we have: 4+3 = 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1.

Alois P. Heinz, Rows n = 1..200, flattened

Column k=1 gives: A025065(n-1) for n>1.

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i=1, 1+x,
       b(n, i-1) +`if`(i>n, 0, (p->`if`(irem(i, 2, 'r')=0, p,
       coeff(p, x, 0)*(1+x^(r+1)) +add(coeff(p, x, j)*x^j,
       j=r+2..degree(p))))(b(n-i, i)))))
    end:
T:= n->(p-> seq(coeff(p, x, j), j=1..degree(p)))(b(n, n)):
seq(T(n), n=1..20);  # Alois P. Heinz, Sep 28 2013

nn=16;Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[Series[x^(2k-1)/Product[1-x^j,{j,1,2k-1}] /Product[(1-x^(2j)),{j,k,nn}],{x,0,nn}],x],{k,1,nn/2}]],1]]//Grid

O.g.f. for column k: x^(2k-1)/[ prod_{j=1..2k-1}(1-x^j)*prod_{j>=k} (1-x^(2j)) ].
For even n=2j and k>=ceiling((n+2)/4) T(n,k)=A058695(j-k).
For odd n=2j-1 and k>=ceiling((n+2)/4) T(n,k)= A058696(j-k).

A322014 Heinz numbers of integer partitions with an even number of even parts.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 11, 16, 17, 18, 20, 21, 22, 23, 25, 31, 32, 34, 36, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 55, 57, 59, 62, 64, 67, 68, 72, 73, 78, 80, 81, 82, 83, 84, 85, 87, 88, 90, 91, 92, 94, 97, 98, 99, 100, 103, 105, 109, 110, 111, 114, 115, 118

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000701, A046682, A056239, A058695, A058696, A108949, A296150, A321753.

a:= proc(n) option remember; local k; for k from 1+`if`(n=1,
      0, a(n-1)) while add(`if`(numtheory[pi](i[1])::odd,
      0, i[2]), i=ifactors(k)[2])::odd do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 24 2018

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],EvenQ[Count[primeMS[#],_?EvenQ]]&]

A341448 Heinz numbers of integer partitions of type OO.

Original entry on oeis.org

6, 14, 15, 24, 26, 33, 35, 38, 51, 54, 56, 58, 60, 65, 69, 74, 77, 86, 93, 95, 96, 104, 106, 119, 122, 123, 126, 132, 135, 140, 141, 142, 143, 145, 150, 152, 158, 161, 177, 178, 185, 201, 202, 204, 209, 214, 215, 216, 217, 219, 221, 224, 226, 232, 234, 240

Offset: 1

Gus Wiseman, Feb 15 2021

nonn

These partitions are defined to have an odd number of odd parts and an odd number of even parts. They also have even length and odd sum.

			The sequence of partitions together with their Heinz numbers begins:
      6: (2,1)         74: (12,1)           141: (15,2)
     14: (4,1)         77: (5,4)            142: (20,1)
     15: (3,2)         86: (14,1)           143: (6,5)
     24: (2,1,1,1)     93: (11,2)           145: (10,3)
     26: (6,1)         95: (8,3)            150: (3,3,2,1)
     33: (5,2)         96: (2,1,1,1,1,1)    152: (8,1,1,1)
     35: (4,3)        104: (6,1,1,1)        158: (22,1)
     38: (8,1)        106: (16,1)           161: (9,4)
     51: (7,2)        119: (7,4)            177: (17,2)
     54: (2,2,2,1)    122: (18,1)           178: (24,1)
     56: (4,1,1,1)    123: (13,2)           185: (12,3)
     58: (10,1)       126: (4,2,2,1)        201: (19,2)
     60: (3,2,1,1)    132: (5,2,1,1)        202: (26,1)
     65: (6,3)        135: (3,2,2,2)        204: (7,2,1,1)
     69: (9,2)        140: (4,3,1,1)        209: (8,5)

Note: A-numbers of ranking sequences are in parentheses below.
The case of odd parts, length, and sum is counted by A078408 (A300272).
The type EE version is A236913 (A340784).
These partitions (for odd n) are counted by A236914.
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd (A340932).
A027193 counts partitions of odd length/maximum (A026424/A244991).
A058695 counts partitions of odd numbers (A300063).
A160786 counts odd-length partitions of odd numbers (A340931).
A340101 counts factorizations into odd factors.
A340385 counts partitions of odd length and maximum (A340386).
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
Cf. A000700, A024429, A027187, A106529, A117409, A174725, A257541, A325134, A339890, A340102, A340604.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[Count[primeMS[#],?EvenQ]]&&OddQ[Count[primeMS[#],?OddQ]]&]

A374921 Irregular triangle read by rows: T(n,k), n >= 0, k >= 1, in which if n is even then row n lists the first A008619(n) even indexed terms of A027336 otherwise if n is odd then row n lists the first A008619(n) odd indexed terms of A027336.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 4, 1, 1, 3, 6, 1, 2, 4, 8, 1, 1, 3, 6, 11, 1, 2, 4, 8, 15, 1, 1, 3, 6, 11, 20, 1, 2, 4, 8, 15, 26, 1, 1, 3, 6, 11, 20, 35, 1, 2, 4, 8, 15, 26, 45, 1, 1, 3, 6, 11, 20, 35, 58, 1, 2, 4, 8, 15, 26, 45, 75, 1, 1, 3, 6, 11, 20, 35, 58, 96, 1, 2, 4, 8, 15, 26, 45, 75, 121

Offset: 0

Omar E. Pol, Aug 01 2024

The sum of row n equals the number of partitions of n.

			Triangle begins:
  1;
  1;
  1, 1;
  1, 2;
  1, 1, 3;
  1, 2, 4;
  1, 1, 3, 6;
  1, 2, 4, 8;
  1, 1, 3, 6, 11;
  1, 2, 4, 8, 15;
  1, 1, 3, 6, 11, 20;
  1, 2, 4, 8, 15, 26;
  1, 1, 3, 6, 11, 20, 35;
  1, 2, 4, 8, 15, 26, 45;
  1, 1, 3, 6, 11, 20, 35, 58;
  1, 2, 4, 8, 15, 26, 45, 75;
  1, 1, 3, 6, 11, 20, 35, 58, 96;
  1, 2, 4, 8, 15, 26, 45, 75, 121;
  ...
For n = 10 the sum of the 10th row is 1 + 1 + 3 + 6 + 11 + 20 = 42, the same as the number of partitions of 10.

Row sums give A000041.
Row lengths give A008619.
Right border gives A027336.
Columns 1..4: A000012, A000034, A010702, A010724.
Cf. A058695, A058696, A002865, A167430, A176206, A182844, A182845, A336811, A338156.

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A371840 Number of integer partitions of n with non-biquanimous multiplicities.

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A342082 Numbers with an inferior odd divisor > 1.

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A111295 Number of partitions of 3n+1.

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A340929 Heinz numbers of integer partitions of odd negative rank.

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A346634 Number of strict odd-length integer partitions of 2n + 1.

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A229724 Triangular array read by rows: T(n,k) is the number of partitions of n in which the greatest odd part is equal to 2k-1; n >= 1, 1 <= k <= ceiling(n/2).

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A322014 Heinz numbers of integer partitions with an even number of even parts.

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A341448 Heinz numbers of integer partitions of type OO.