A066208 -id:A066208 - cp's OEIS frontend

A244991 Numbers whose greatest prime factor is a prime with an odd index; n such that A006530(n) is in A031368.

2, 4, 5, 8, 10, 11, 15, 16, 17, 20, 22, 23, 25, 30, 31, 32, 33, 34, 40, 41, 44, 45, 46, 47, 50, 51, 55, 59, 60, 62, 64, 66, 67, 68, 69, 73, 75, 77, 80, 82, 83, 85, 88, 90, 92, 93, 94, 97, 99, 100, 102, 103, 109, 110, 115, 118, 119, 120, 121, 123, 124, 125, 127, 128

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

Equally, numbers n for which A061395(n) is odd.
A122111 maps each one of these numbers to a unique term of A026424 and vice versa.
If the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), these are the Heinz numbers of partitions whose greatest part is odd, counted by A027193. - Gus Wiseman, Feb 08 2021

			From _Gus Wiseman_, Feb 08 2021: (Start)
The sequence of terms together with their prime indices begins:
      2: {1}           32: {1,1,1,1,1}     64: {1,1,1,1,1,1}
      4: {1,1}         33: {2,5}           66: {1,2,5}
      5: {3}           34: {1,7}           67: {19}
      8: {1,1,1}       40: {1,1,1,3}       68: {1,1,7}
     10: {1,3}         41: {13}            69: {2,9}
     11: {5}           44: {1,1,5}         73: {21}
     15: {2,3}         45: {2,2,3}         75: {2,3,3}
     16: {1,1,1,1}     46: {1,9}           77: {4,5}
     17: {7}           47: {15}            80: {1,1,1,1,3}
     20: {1,1,3}       50: {1,3,3}         82: {1,13}
     22: {1,5}         51: {2,7}           83: {23}
     23: {9}           55: {3,5}           85: {3,7}
     25: {3,3}         59: {17}            88: {1,1,1,5}
     30: {1,2,3}       60: {1,1,2,3}       90: {1,2,2,3}
     31: {11}          62: {1,11}          92: {1,1,9}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10001

Complement: A244990.
Cf. A006530, A026424, A031368, A122111, A244321, A244322, A244989.
Looking at least instead of greatest prime index gives A026804.
The partitions with these Heinz numbers are counted by A027193.
The case where Omega is odd also is A340386.
A001222 counts prime factors.
A056239 adds up prime indices.
A300063 ranks partitions of odd numbers.
A061395 selects maximum prime index.
A066208 ranks partitions into odd parts.
A112798 lists the prime indices of each positive integer.
A340931 ranks odd-length partitions of odd numbers.
Cf. A000009, A058695, A072233, A160786, A300272, A340101, A340385, A340604.

Select[Range[100],OddQ[PrimePi[FactorInteger[#][[-1,1]]]]&] (* Gus Wiseman, Feb 08 2021 *)

For all n, A244989(a(n)) = n.

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

Original entry on oeis.org

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]

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

A160786 The number of odd partitions of consecutive odd integers.

Original entry on oeis.org

1, 2, 4, 8, 16, 29, 52, 90, 151, 248, 400, 632, 985, 1512, 2291, 3431, 5084, 7456, 10836, 15613, 22316, 31659, 44601, 62416, 86809, 120025, 165028, 225710, 307161, 416006, 560864, 752877, 1006426, 1340012, 1777365, 2348821, 3093095, 4059416, 5310255, 6924691

Offset: 0

Utpal Sarkar (doetoe(AT)gmail.com), May 26 2009

nonn

It seems that these are partitions of odd length and sum, ranked by A340931. The parts do not have to be odd. - Gus Wiseman, Apr 06 2021

			From _Gus Wiseman_, Apr 06 2021: (Start)
The a(0) = 1 through a(4) = 16 partitions:
  (1)  (3)    (5)      (7)        (9)
       (111)  (221)    (322)      (333)
              (311)    (331)      (432)
              (11111)  (421)      (441)
                       (511)      (522)
                       (22111)    (531)
                       (31111)    (621)
                       (1111111)  (711)
                                  (22221)
                                  (32211)
                                  (33111)
                                  (42111)
                                  (51111)
                                  (2211111)
                                  (3111111)
                                  (111111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..5000
V. I. Arnold, On teaching mathematics

Partitions with all odd parts are counted by A000009 and ranked by A066208.
This is a bisection of A027193 (odd-length partitions), which is ranked by A026424.
The case of all odd parts is counted by A078408 and ranked by A300272.
The even version is A236913, ranked by A340784.
A multiplicative version is A340102.
These partitions are ranked by A340931.
A047993 counts balanced partitions, ranked by A106529.
A058695 counts partitions of odd numbers, ranked by A300063.
A072233 counts partitions by sum and length.
A236914 counts partition of type OO, ranked by A341448.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Cf. A000700, A168659, A200750, A340604, A340607, A340854, A340855.

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],
      `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->
      `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],
          [p[2], p[1], p[4], p[3]]))(b(n-i, i)))))
    end:
a:= n-> b(2*n+1$2)[2]:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 16 2014

b[n_, i_] := b[n, i] = If[n==0, {1, 0, 0, 0}, If[i<1, {0, 0, 0, 0}, b[n, i-1] + If[i>n, {0, 0, 0, 0}, Function[{p}, If[Mod[i, 2]==0, p[[{3, 4, 1, 2}]], p[[{2, 1, 4, 3}]]]][b[n-i, i]]]]]; a[n_] := b[2*n+1, 2*n+1][[2]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 01 2015, after Alois P. Heinz *)
(* Slow but easy to read *)
a[n_] := Length@IntegerPartitions[2 n + 1, {1, 2 n + 1, 2}]
a /@ Range[0, 25]
(* Leo C. Stein, Nov 11 2020 *)
(* Faster, don't build the partitions themselves *)
(* Number of partitions of n into exactly k parts *)
P[0, 0] = 1;
P[n_, k_] := 0 /; ((k <= 0) || (n <= 0))
P[n_, k_] := P[n, k] = P[n - k, k] + P[n - 1, k - 1]
a[n_] := Sum[P[2 n + 1, k], {k, 1, 2 n + 1, 2}]
a /@ Range[0, 40]
(* Leo C. Stein, Nov 11 2020 *)

# Could be memoized for speedup
def numoddpart(n, m=1):
    """The number of partitions of n into an odd number of parts of size at least m"""
    if n < m:
        return 0
    elif n == m:
        return 1
    else:
        # 1 (namely n = n) and all partitions of the form
        # k + even partitions that start with >= k
        return 1 + sum([numevenpart(n - k,  k) for k in range(m, n//3 + 1)])
def numevenpart(n, m=1):
    """The number of partitions of n into an even number of parts of size at least m"""
    if n < 2*m:
        return 0
    elif n == 2*m:
        return 1
    else:
        return sum([numoddpart(n - k,  k) for k in range(m,  n//2 + 1)])
[numoddpart(n) for n in range(1, 70, 2)]

# dict to memoize
ps = {(0,0): 1}
def p(n, k):
    """Number of partitions of n into exactly k parts"""
    if (n,k) in ps: return ps[(n,k)]
    if (n<=0) or (k<=0): return 0
    ps[(n,k)] = p(n-k,k) + p(n-1,k-1)
    return ps[(n,k)]
def a(n): return sum([p(2*n+1, k) for k in range(1,2*n+3,2)])
[a(n) for n in range(0,41)]
# Leo C. Stein, Nov 11 2020

a(n) = A027193(2n+1).

A300272 Sorted list of Heinz numbers of odd partitions.

Original entry on oeis.org

2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 50, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 110, 124, 125, 127, 128, 137, 149, 157, 164, 167, 170, 176, 179, 188, 191, 197, 200, 211, 227, 230, 233, 236, 241, 242, 257, 268, 269, 272, 275, 277, 283, 292, 307, 310

Offset: 1

Gus Wiseman, Mar 01 2018

nonn

An odd partition is an integer partition of an odd number into an odd number of parts, all of which are odd.

Any product of three members of this sequence is also in the sequence.

			Sequence of odd partitions begins: (1), (3), (111), (5), (7), (311), (9), (11), (11111), (13), (511), (15), (331), (17), (19), (711), (21), (31111), (23), (911), (25), (27), (29), (531), (1111), (333), (31), (1111111).

Cf. A000009, A031368, A066208, A078408, A094457, A215366, A299757, A300063.

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

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.

A366528 Sum of odd prime indices of n.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 0, 3, 0, 4, 5, 2, 0, 1, 3, 4, 7, 1, 0, 5, 0, 6, 9, 3, 6, 1, 0, 2, 0, 4, 11, 5, 5, 8, 3, 2, 0, 1, 0, 6, 13, 1, 0, 7, 3, 10, 15, 4, 0, 7, 7, 2, 0, 1, 8, 3, 0, 1, 17, 5, 0, 12, 0, 6, 3, 6, 19, 9, 9, 4, 0, 3, 21, 1, 6, 2, 5, 1, 0, 7, 0, 14, 23, 2

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, sum A056239(n).

			The prime indices of 198 are {1,2,2,5}, so a(198) = 1+5 = 6.

Zeros are A066207, counted by A035363.
The triangle for this rank statistic is A113685, without zeros A365067.
For count instead of sum we have A257991, even A257992.
Nonzeros are A366322, counted by A086543.
The even version is A366531, halved A366533, triangle A113686.
A000009 counts partitions into odd parts, ranks A066208.
A053253 = partitions with all odd parts and conjugate parts, ranks A352143.
A066967 adds up sums of odd parts over all partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A055396, A130780, A171966, A174713, A325698, A325700, A366748.

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

a(n) = A056239(n) - A366531(n).

A340101 Number of factorizations of 2n + 1 into odd factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 4, 1, 2, 2, 1, 2, 2, 1, 1, 4, 2, 1, 2, 1, 1, 4, 2, 1, 5, 1, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 5, 1, 1, 2, 1, 2, 4, 2, 2, 2, 3, 1, 2, 1, 2, 7, 1, 1, 2, 2, 2, 4, 1, 1, 4, 2, 1, 2, 2, 1, 5, 1, 2, 4, 1, 4, 2, 1, 1, 2, 2, 2, 7, 1, 1, 5, 1, 1, 2, 2, 2, 4, 2

Offset: 0

Gus Wiseman, Dec 28 2020

nonn

			The factorizations for 2n + 1 = 27, 45, 135, 225, 315, 405, 1155:
  27      45      135       225       315       405         1155
  3*9     5*9     3*45      3*75      5*63      5*81        15*77
  3*3*3   3*15    5*27      5*45      7*45      9*45        21*55
          3*3*5   9*15      9*25      9*35      15*27       33*35
                  3*5*9     15*15     15*21     3*135       3*385
                  3*3*15    5*5*9     3*105     5*9*9       5*231
                  3*3*3*5   3*3*25    5*7*9     3*3*45      7*165
                            3*5*15    3*3*35    3*5*27      11*105
                            3*3*5*5   3*5*21    3*9*15      3*5*77
                                      3*7*15    3*3*5*9     3*7*55
                                      3*3*5*7   3*3*3*15    5*7*33
                                                3*3*3*3*5   3*11*35
                                                            5*11*21
                                                            7*11*15
                                                            3*5*7*11

Antti Karttunen, Table of n, a(n) for n = 0..32768

The version for partitions is A160786, ranked by A300272.
The even version is A340785.
The odd-length case is A340102.
A000009 counts partitions into odd parts, ranked by A066208.
A001055 counts factorizations, with strict case A045778.
A027193 counts partitions of odd length, ranked by A026424.
A058695 counts partitions of odd numbers, ranked by A300063.
A316439 counts factorizations by product and length.
Cf. A000700, A002033, A027187, A028260, A074206, A078408, A174726, A236914, A320732, A339846.
Odd bisection of A001055, and also of A349907.

g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> g(2*n+1$2):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ[Times@@#]&]],{n,1,100,2}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s)); \\ After code in A001055
A340101(n) = A001055(n+n+1); \\ Antti Karttunen, Dec 13 2021

a(n) = A001055(2n+1).

a(n) = A349907(2n+1). - Antti Karttunen, Dec 13 2021

Data section extended up to 105 terms by Antti Karttunen, Dec 13 2021

A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.

Original entry on oeis.org

1, 4, 6, 15, 16, 21, 24, 25, 35, 60, 64, 77, 84, 90, 91, 96, 100, 121, 126, 140, 143, 150, 210, 221, 240, 247, 256, 289, 297, 308, 323, 336, 351, 360, 364, 375, 384, 400, 437, 462, 484, 490, 495, 504, 525, 529, 546, 551, 560, 572, 585, 600, 625, 667, 686, 726

Offset: 1

Gus Wiseman, Jan 21 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with the same number of even prime indices as odd conjugate prime indices.

These are also partitions for which the number of even parts is equal to the positive alternating sum of the parts.

			The terms and their prime indices begin:
    1: ()
    4: (1,1)
    6: (2,1)
   15: (3,2)
   16: (1,1,1,1)
   21: (4,2)
   24: (2,1,1,1)
   25: (3,3)
   35: (4,3)
   60: (3,2,1,1)
   64: (1,1,1,1,1,1)
   77: (5,4)
   84: (4,2,1,1)
   90: (3,2,2,1)
   91: (6,4)
   96: (2,1,1,1,1,1)

A subset of A028260 (even bigomega), counted by A027187.
These partitions are counted by A277579.
This is the half-conjugate version of A325698, counted by A045931.
A000041 counts partitions, strict A000009.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A100824 counts partitions with at most one odd part, ranked by A349150.
A108950/A108949 count partitions with more odd/even parts.
A122111 represents conjugation using Heinz numbers.
A130780/A171966 count partitions with more or equal odd/even parts.
A257991/A257992 count odd/even prime indices.
A316524 gives the alternating sum of prime indices (reverse: A344616).
Cf. A000700, A000712, A035363, A066207, A066208, A097613, A215366, A239241, A240009, A241638, A316523, A325700, A340604.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[primeMS[#],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

A257992(a(n)) = A257991(A122111(a(n))).

A367224 Numbers m with a divisor whose prime indices sum to bigomega(m).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 30, 32, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 56, 57, 60, 64, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 90, 93, 96, 100, 102, 105, 108, 110, 111, 112, 114, 120, 123, 125, 126, 128, 129, 130, 132, 135, 138, 140, 141

Offset: 1

Gus Wiseman, Nov 14 2023

nonn

Also numbers m whose prime indices have a submultiset summing to bigomega(m).
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.
These are the Heinz numbers of the partitions counted by A367212.

			The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224*   A367225    A367226    A367227
A000700 counts self-conjugate partitions, ranks A088902.
A002865 counts partitions whose length is a part, ranks A325761.
A005117 ranks strict integer partitions, counted by A000009.
A066208 ranks partitions into odd parts, also counted by A000009.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A126796 counts complete partitions, ranks A325781.
A229816 counts partitions whose length is not a part, ranks A367107.
A237668 counts sum-full partitions, ranks A364532.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365543 counts partitions of n with a subset-sum k, strict A365661.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A000720, A055396, A061395, A106529, A299729, A304792, A363225, A365830.

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], MemberQ[Total/@prix/@Divisors[#], PrimeOmega[#]]&]

cp's OEIS Frontend

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