A006141 -id:A006141 - cp's OEIS frontend

A340596 Number of co-balanced factorizations of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 8

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization of n into factors > 1 to be co-balanced if it has exactly A001221(n) factors.

			The a(n) co-balanced factorizations for n = 12, 24, 36, 72, 120, 144, 180:
  2*6    3*8     4*9     8*9     3*5*8     2*72     4*5*9
  3*4    4*6     6*6     2*36    4*5*6     3*48     5*6*6
         2*12    2*18    3*24    2*2*30    4*36     2*2*45
                 3*12    4*18    2*3*20    6*24     2*3*30
                         6*12    2*4*15    8*18     2*5*18
                                 2*5*12    9*16     2*6*15
                                 2*6*10    12*12    2*9*10
                                 3*4*10             3*3*20
                                                    3*4*15
                                                    3*5*12
                                                    3*6*10

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

Positions of terms > 1 are A126706.
Positions of 1's are A303554.
The version for unlabeled multiset partitions is A319616.
The alt-balanced version is A340599.
The balanced version is A340653.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A003963, A006141, A050320, A112798, A117409, A324518, A339846, A339890, A340607, A340656, A340657.

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],Length[#]==PrimeNu[n]&]],{n,100}]

A340596(n, m=n, om=omega(n)) = if(1==n,(0==om), sumdiv(n, d, if((d>1)&&(d<=m), A340596(n/d, d, om-1)))); \\ Antti Karttunen, Jun 10 2024

Data section extended up to a(120) by Antti Karttunen, Jun 10 2024

A039900 Number of partitions satisfying 0 < cn(0,5) + cn(1,5) + cn(4,5).

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 9, 13, 19, 27, 38, 52, 71, 95, 127, 167, 220, 285, 370, 474, 607, 770, 976, 1226, 1540, 1920, 2391, 2960, 3660, 4501, 5529, 6760, 8254, 10038, 12190, 14750, 17825, 21470, 25825, 30975, 37101, 44322, 52879, 62937, 74811, 88733, 105110, 124261

Offset: 0

Olivier Gérard

nonn

For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: o < 0 + 1 + 4 (OMZAAp).
Number of partitions of n such that (greatest part) >= (multiplicity of greatest part), for n >= 1.  For example, a(6) counts these 9 partitions:  6, 51, 42, 411, 33, 321, 3111, 22111, 21111.  See the Mathematica program at A240057 for the sequence as a count of these partitions, along with counts of related partitions.  - Clark Kimberling, Apr 02 2014
The Heinz numbers of these integer partitions are given by A324561. - Gus Wiseman, Mar 09 2019
From Gus Wiseman, Mar 09 2019: (Start)
Also the number of integer partitions of n whose minimum part is less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324560. For example, the a(1) = 1 through a(7) = 13 integer partitions are:
  (1)  (11)  (21)   (22)    (32)     (42)      (52)
             (111)  (31)    (41)     (51)      (61)
                    (211)   (221)    (222)     (322)
                    (1111)  (311)    (321)     (331)
                            (2111)   (411)     (421)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

			From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(7) = 13 integer partitions with at least one part equal to 0, 1, or 4 modulo 5:
  (1)  (11)  (21)   (4)     (5)      (6)       (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (321)     (331)
                            (2111)   (411)     (421)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A003106, A003114, A039899.
Cf. A003114, A006141, A047993, A064174, A117144.
Cf. A324518, A324520, A324522, A324560, A324561.

b:= proc(n, i, t) option remember; `if`(n=0, t,
      `if`(i<1, 0, b(n, i-1, t)+ `if`(i>n, 0, b(n-i, i,
      `if`(irem(i, 5) in {2, 3}, t, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 03 2014

Table[Count[IntegerPartitions[n], p_ /; Min[p] <= Length[p]], {n, 40}] (* Clark Kimberling, Feb 13 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0, t, If[i<1, 0, b[n, i-1, t] + If[i > n, 0, b[n-i, i, If[MemberQ[{2, 3}, Mod[i, 5]], t, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, x^k*(1-x^k^2)/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

G.f.: Sum_{k>=0} x^k * (1-x^(k^2)) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

a(n) = A000041(n) - A003106(n). - Vaclav Kotesovec, Oct 20 2024

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.

A340610 Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 11, 13, 14, 17, 19, 20, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 45, 47, 49, 50, 52, 53, 56, 57, 58, 59, 61, 65, 67, 71, 73, 74, 75, 78, 79, 83, 84, 86, 87, 89, 91, 92, 95, 97, 101, 103, 106, 107, 109, 111, 113, 117, 122, 125, 126, 127

Offset: 1

Gus Wiseman, Jan 27 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 sequence of terms together with their prime indices begins:
     2: {1}        29: {10}       56: {1,1,1,4}
     3: {2}        30: {1,2,3}    57: {2,8}
     5: {3}        31: {11}       58: {1,10}
     6: {1,2}      35: {3,4}      59: {17}
     7: {4}        37: {12}       61: {18}
     9: {2,2}      38: {1,8}      65: {3,6}
    11: {5}        39: {2,6}      67: {19}
    13: {6}        41: {13}       71: {20}
    14: {1,4}      43: {14}       73: {21}
    17: {7}        45: {2,2,3}    74: {1,12}
    19: {8}        47: {15}       75: {2,3,3}
    20: {1,1,3}    49: {4,4}      78: {1,2,6}
    21: {2,4}      50: {1,3,3}    79: {22}
    23: {9}        52: {1,1,6}    83: {23}
    26: {1,6}      53: {16}       84: {1,1,2,4}

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

Note: Heinz numbers are given in parentheses below.
The case of equality is A047993 (A106529).
The case where all parts are multiples, not just the maximum part, is A143773 (A316428), with strict case A340830, while the case of factorizations is A340853.
These are the Heinz numbers of certain partitions counted by A168659.
The reciprocal version is A340609.
The squarefree case is A340828 (A340856).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
Cf. A244990/A244991, A326837, A326849 (A326848), A340653, A340691, A340693 (A340606), A340787/A340788.

filter:= proc(n) local F,m,g,t;
  F:= ifactors(n)[2];
  m:= add(t[2],t=F);
  g:= numtheory:-pi(max(seq(t[1],t=F)));
  g mod m = 0;
end proc:
select(filter, [$2..1000]); # Robert Israel, Feb 08 2021

Select[Range[2,100],Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

A001222(a(n)) divides A061395(a(n)).

A340601 Number of integer partitions of n of even rank.

Original entry on oeis.org

1, 1, 0, 3, 1, 5, 3, 11, 8, 18, 16, 34, 33, 57, 59, 98, 105, 159, 179, 262, 297, 414, 478, 653, 761, 1008, 1184, 1544, 1818, 2327, 2750, 3480, 4113, 5137, 6078, 7527, 8899, 10917, 12897, 15715, 18538, 22431, 26430, 31805, 37403, 44766, 52556, 62620, 73379

Offset: 0

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. For this sequence, the rank of an empty partition is 0.

			The a(1) = 1 through a(9) = 18 partitions (empty column indicated by dot):
  (1)  .  (3)    (22)  (5)      (42)    (7)        (44)      (9)
          (21)         (41)     (321)   (43)       (62)      (63)
          (111)        (311)    (2211)  (61)       (332)     (81)
                       (2111)           (322)      (521)     (333)
                       (11111)          (331)      (2222)    (522)
                                        (511)      (4211)    (531)
                                        (2221)     (32111)   (711)
                                        (4111)     (221111)  (4221)
                                        (31111)              (4311)
                                        (211111)             (6111)
                                        (1111111)            (32211)
                                                             (33111)
                                                             (51111)
                                                             (222111)
                                                             (411111)
                                                             (3111111)
                                                             (21111111)
                                                             (111111111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
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: Heinz numbers are given in parentheses below.
The positive case is A101708 (A340605).
The Heinz numbers of these partitions are A340602.
The odd version is A340692 (A340603).
- Rank -
A047993 counts partitions of rank 0 (A106529).
A072233 counts partitions by sum and length.
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts factorizations of rank 0.
- Even -
A024430 counts set partitions of even length.
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A052841 counts ordered set partitions of even length.
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts even-length partitions of even numbers (A340784).
A339846 counts factorizations of even length.
Cf. A000041, A006141, A039900, A064174, A067538, A117409, A200750, A324516.

b:= proc(n, i, r) option remember; `if`(n=0, 1-max(0, r),
      `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
      `if`(r<0, irem(i, 2), r))))
    end:
a:= n-> b(n$2, -1):
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 22 2021

Table[If[n==0,1,Length[Select[IntegerPartitions[n],EvenQ[Max[#]-Length[#]]&]]],{n,0,30}]
(* Second program: *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, 1 - Max[0, r], If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 - If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1];
a /@ Range[0, 55] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

p_q(k) = {prod(j=1, k, 1-q^j); }
GB_q(N, M)= {if(N>=0 && M>=0,  p_q(N+M)/(p_q(M)*p_q(N)), 0 ); }
A_q(N) = {my(q='q+O('q^N), g=1+sum(i=1,N, sum(j=1,N/i, q^(i*j) * ( ((1/2)*(1+(-1)^(i+j))) + sum(k=1,N-(i*j), ((q^k)*GB_q(k,i-2)) * ((1/2)*(1+(-1)^(i+j+k)))))))); Vec(g)}
A_q(50) \\ John Tyler Rascoe, Apr 15 2024

G.f.: 1 + Sum_{i, j>0} q^(i*j) * ( (1+(-1)^(i+j))/2 + Sum_{k>0} q^k * q_binomial(k,i-2) * (1+(-1)^(i+j+k))/2 ). - John Tyler Rascoe, Apr 15 2024

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - Vaclav Kotesovec, Apr 17 2024

A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Jan 20 2021

nonn

I call these alt-balanced factorizations. Balanced factorizations are A340653. - Gus Wiseman, Jan 20 2021

			The alt-balanced factorizations for n = 192, 1728, 3456, 9216:
  3*4*4*4       2*2*2*6*6*6   2*2*4*6*6*6         4*4*4*4*6*6
  2*2*2*2*2*6   2*2*3*4*6*6   2*3*4*4*6*6         2*2*2*2*2*6*6*8
                2*3*3*4*4*6   3*3*4*4*4*6         2*2*2*2*3*3*8*8
                              2*2*2*2*3*3*3*8     2*2*2*2*3*4*6*8
                              2*2*2*2*2*2*2*3*9   2*2*2*3*3*4*4*8
                                                  2*2*2*2*2*2*2*8*9
                                                  2*2*2*2*2*2*4*4*9

Antti Karttunen, Table of n, a(n) for n = 1..100000

The co-balanced version is A340596.
Positions of nonzero terms are A340597.
The case of powers of two is A340611.
Taking maximum Omega instead of maximum factor gives A340653.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A006141, A117409, A320655, A320656, A324518, A339846, A339890, A340607.

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],Length[#]==Max[#]&]],{n,100}]

A340599(n, m=n, e=0, mf=1) = if(1==n, mf==e, sumdiv(n, d, if((d>1)&&(d<=m), A340599(n/d, d, 1+e, max(d, mf))))); \\ Antti Karttunen, Jun 19 2024

Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Jun 19 2024

A117144 Partitions of n in which each part k occurs at least k times.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 12, 15, 16, 19, 21, 25, 28, 32, 34, 41, 46, 51, 55, 64, 70, 79, 86, 97, 106, 119, 129, 146, 159, 175, 190, 214, 232, 256, 277, 306, 334, 367, 394, 434, 472, 515, 556, 607, 654, 714, 770, 836, 901, 978, 1048, 1140, 1226, 1322

Offset: 0

Emeric Deutsch, Mar 06 2006

nonn

The Heinz numbers of these integer partitions are given by A324525. - Gus Wiseman, Mar 09 2019

			a(9)=5 because we have [3,3,3], [2,2,2,2,1], [2,2,2,1,1,1], [2,2,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(9) = 5 integer partitions:
  1  11  111  22    221    222     2221     2222      333
              1111  11111  2211    22111    22211     22221
                           111111  1111111  221111    222111
                                            11111111  2211111
                                                      111111111
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001462, A003114, A006141, A033461, A039900, A047993, A052335, A064174, A090858, A114638, A115584, A276078, A280204.

Cf. A324518, A324520, A324525, A324572.

g:=product((1-x^k+x^(k^2))/(1-x^k),k=1..100): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..66);
# second Maple program:
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +add(b(n-i*j, i-1), j=i..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 28 2016

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1], {j, i, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>=i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Mar 09 2019 *)
nmax = 100; CoefficientList[Series[Product[(1-x^k+x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 28 2024 *)

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

A340609 Numbers whose number of prime factors (A001222) is divisible by their greatest prime index (A061395).

Original entry on oeis.org

2, 4, 6, 8, 9, 16, 20, 24, 30, 32, 36, 45, 50, 54, 56, 64, 75, 81, 84, 96, 125, 126, 128, 140, 144, 160, 176, 189, 196, 210, 216, 240, 256, 264, 294, 315, 324, 350, 360, 384, 396, 400, 416, 440, 441, 486, 490, 512, 525, 540, 576, 594, 600, 616, 624, 660, 686

Offset: 1

Gus Wiseman, Jan 27 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.

If n is a term, then so is n^k for k > 1. - Robert Israel, Feb 08 2021

			The sequence of terms together with their prime indices begins:
      2: {1}             64: {1,1,1,1,1,1}      216: {1,1,1,2,2,2}
      4: {1,1}           75: {2,3,3}            240: {1,1,1,1,2,3}
      6: {1,2}           81: {2,2,2,2}          256: {1,1,1,1,1,1,1,1}
      8: {1,1,1}         84: {1,1,2,4}          264: {1,1,1,2,5}
      9: {2,2}           96: {1,1,1,1,1,2}      294: {1,2,4,4}
     16: {1,1,1,1}      125: {3,3,3}            315: {2,2,3,4}
     20: {1,1,3}        126: {1,2,2,4}          324: {1,1,2,2,2,2}
     24: {1,1,1,2}      128: {1,1,1,1,1,1,1}    350: {1,3,3,4}
     30: {1,2,3}        140: {1,1,3,4}          360: {1,1,1,2,2,3}
     32: {1,1,1,1,1}    144: {1,1,1,1,2,2}      384: {1,1,1,1,1,1,1,2}
     36: {1,1,2,2}      160: {1,1,1,1,1,3}      396: {1,1,2,2,5}
     45: {2,2,3}        176: {1,1,1,1,5}        400: {1,1,1,1,3,3}
     50: {1,3,3}        189: {2,2,2,4}          416: {1,1,1,1,1,6}
     54: {1,2,2,2}      196: {1,1,4,4}          440: {1,1,1,3,5}
     56: {1,1,1,4}      210: {1,2,3,4}          441: {2,2,4,4}

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

Note: Heinz numbers are given in parentheses below.
The case of equality is A047993 (A106529).
These are the Heinz numbers of certain partitions counted by A168659.
The reciprocal version is A340610, with strict case A340828 (A340856).
If all parts (not just the greatest) are divisors we get A340693 (A340606).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
Cf. A039900, A064174, A143773, A244990/A244991, A326837, A326849 (A326848), A340653, A340787/A340788.

filter:= proc(n) local F,m,g,t;
  F:= ifactors(n)[2];
  m:= add(t[2],t=F);
  g:= numtheory:-pi(max(seq(t[1],t=F)));
  m mod g = 0;
end proc:
seelect(filter, [$2..1000]); # Robert Israel, Feb 08 2021

Select[Range[2,100],Divisible[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]&]

A061395(a(n)) divides A001222(a(n)).

A114638 Number of partitions of n such that number of parts is equal to the sum of parts counted without multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 1, 0, 2, 2, 3, 5, 5, 6, 9, 7, 8, 14, 12, 16, 21, 28, 32, 43, 47, 61, 68, 84, 89, 109, 126, 140, 170, 198, 227, 261, 323, 362, 427, 501, 581, 658, 794, 880, 1036, 1175, 1355, 1526, 1776, 1985, 2281, 2588, 2943, 3312, 3799, 4271, 4852, 5497

Offset: 0

Vladeta Jovovic, Feb 18 2006

nonn

The Heinz numbers of these integer partitions are given by A324570. - Gus Wiseman, Mar 09 2019

			a(10) = 3 because we have [5,1,1,1,1,1], [3,3,3,1] and [3,2,2,1,1,1].
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(12) = 5 integer partitions (empty columns not shown):
  1  22   221  3111  3311   333     3331    32222    33222
     211             41111  321111  322111  44111    322221
                                    511111  322211   332211
                                            332111   4221111
                                            4211111  6111111
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A003114, A006141, A039900, A047993, A064174, A066328, A243149 (the same for compositions).
Cf. A324516, A324518, A324520, A324521, A324570.
Cf. A116861 (number of partitions of n having a given sum of distinct parts).

a:=proc(n) local P,c,j,S: with(combinat): P:=partition(n): c:=0: for j from 1 to nops(P) do S:=convert(P[j],set): if nops(P[j])=sum(S[i],i=1..nops(S)) then c:=c+1 else c:=c fi: c: od: end: seq(a(n), n=0..35); # Emeric Deutsch, Mar 01 2006

a[n_] := Module[{P, c, j, S}, P = IntegerPartitions[n]; c = 0; For[j = 1, j <= Length[P], j++, S = Union[P[[j]]]; If[Length[P[[j]]] == Total[S],  c++] ]; c];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 07 2018, after Emeric Deutsch *)

apply( A114638(n,s=0)={forpart(p=n,#p==vecsum(Set(p))&&s++); s}, [0..50]) \\ M. F. Hasler, Oct 27 2019

More terms from Emeric Deutsch, Mar 01 2006

A341446 Heinz numbers of integer partitions whose only odd part is the smallest.

Original entry on oeis.org

2, 5, 6, 11, 14, 17, 18, 23, 26, 31, 35, 38, 41, 42, 47, 54, 58, 59, 65, 67, 73, 74, 78, 83, 86, 95, 97, 98, 103, 106, 109, 114, 122, 126, 127, 137, 142, 143, 145, 149, 157, 158, 162, 167, 174, 178, 179, 182, 185, 191, 197, 202, 209, 211, 214, 215, 222, 226

Offset: 1

Gus Wiseman, Feb 12 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only odd prime index (counting multiplicity) is the smallest.

			The sequence of partitions together with their Heinz numbers begins:
      2: (1)         54: (2,2,2,1)    109: (29)
      5: (3)         58: (10,1)       114: (8,2,1)
      6: (2,1)       59: (17)         122: (18,1)
     11: (5)         65: (6,3)        126: (4,2,2,1)
     14: (4,1)       67: (19)         127: (31)
     17: (7)         73: (21)         137: (33)
     18: (2,2,1)     74: (12,1)       142: (20,1)
     23: (9)         78: (6,2,1)      143: (6,5)
     26: (6,1)       83: (23)         145: (10,3)
     31: (11)        86: (14,1)       149: (35)
     35: (4,3)       95: (8,3)        157: (37)
     38: (8,1)       97: (25)         158: (22,1)
     41: (13)        98: (4,4,1)      162: (2,2,2,2,1)
     42: (4,2,1)    103: (27)         167: (39)
     47: (15)       106: (16,1)       174: (10,2,1)

These partitions are counted by A035363 (shifted left once).
Terms of A340932 can be factored into elements of this sequence.
The even version is A341447.
A001222 counts prime factors.
A005408 lists odd numbers.
A026804 counts partitions whose smallest part is odd.
A027193 counts odd-length partitions, ranked by A026424.
A031368 lists odd-indexed primes.
A032742 selects largest proper divisor.
A055396 selects smallest prime index.
A056239 adds up prime indices.
A058695 counts partitions of odd numbers, ranked by A300063.
A061395 selects largest prime index.
A066207 lists numbers with all even prime indices.
A066208 lists numbers with all odd prime indices.
A112798 lists the prime indices of each positive integer.
A244991 lists numbers whose greatest prime index is odd.
A340932 lists numbers whose smallest prime index is odd.
Cf. A006141, A160786, A257991, A257992, A300272, A340604, A340607, A340854/A340855, A340933.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],OddQ[First[primeMS[#]]]&&And@@EvenQ[Rest[primeMS[#]]]&]

Also numbers n > 1 such that A055396(n) is odd and A032742(n) belongs to A066207.

cp's OEIS Frontend

A340596 Number of co-balanced factorizations of n.

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A039900 Number of partitions satisfying 0 < cn(0,5) + cn(1,5) + cn(4,5).

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A340604 Heinz numbers of integer partitions of odd positive rank.

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A340610 Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).

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A340601 Number of integer partitions of n of even rank.

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A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.

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A117144 Partitions of n in which each part k occurs at least k times.

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