A344416 -id:A344416 - cp's OEIS frontend

A025065 Number of palindromic partitions of n.

1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296, 11732, 11732, 14742, 14742, 18460, 18460, 23025, 23025, 28629, 28629

Offset: 0

Clark Kimberling

nonn

That is, the number of partitions of n into parts which can be listed in palindromic order.
Alternatively, number of partitions of n into parts from the set {1,2,4,6,8,10,12,...}. - T. D. Noe, Aug 05 2005
Also, partial sums of A035363.
Also number of partitions of n with at most one part occurring an odd number of times. - Reinhard Zumkeller, Dec 18 2013
The first Mathematica program computes terms of A025065; the second computes the k palindromic partitions of user-chosen n. - Clark Kimberling, Jan 20 2014
a(n) is the number of partitions p of n+1 such that 2*max(p) > n+1. - Clark Kimberling, Apr 20 2014.
From Gus Wiseman, Nov 28 2018: (Start)
Also the number of integer partitions of n + 2 that are the vertex-degrees of some hypertree. For example, the a(6) = 7 partitions of 8 that are the vertex-degrees of some hypertree, together with a realizing hypertree are:
     (41111): {{1,2},{1,3},{1,4},{1,5}}
     (32111): {{1,2},{1,3},{1,4},{2,5}}
     (22211): {{1,2},{1,3},{2,4},{3,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
    (221111): {{1,2},{1,3},{2,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}
(End)
Conjecture: a(n) is the length of maximal initial segment of A308355(n-1) that is identical to row n of A128628, for n >= 2. - Clark Kimberling, May 24 2019
From Gus Wiseman, May 21 2021: (Start)
The Heinz numbers of palindromic partitions are given by A265640. 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.
Also the number of integer partitions of n with a part greater than or equal to n/2. This is equivalent to Clark Kimberling's final comment above. The Heinz numbers of these partitions are given by A344414. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)     (8)
       (11)  (21)  (22)   (32)   (33)    (43)    (44)
                   (31)   (41)   (42)    (52)    (53)
                   (211)  (311)  (51)    (61)    (62)
                                 (321)   (421)   (71)
                                 (411)   (511)   (422)
                                 (3111)  (4111)  (431)
                                                 (521)
                                                 (611)
                                                 (4211)
                                                 (5111)
                                                 (41111)
Also the number of integer partitions of n with at least n/2 parts. The Heinz numbers of these partitions are given by A344296. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)   (21)   (22)    (221)    (222)     (2221)     (2222)
       (11)  (111)  (31)    (311)    (321)     (3211)     (3221)
                    (211)   (2111)   (411)     (4111)     (3311)
                    (1111)  (11111)  (2211)    (22111)    (4211)
                                     (3111)    (31111)    (5111)
                                     (21111)   (211111)   (22211)
                                     (111111)  (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

			The partitions for the first few values of n are as follows:
n: partitions .......................... number
1: 1 ................................... 1
2: 2 11 ................................ 2
3: 3 111 ............................... 2
4: 4 22 121 1111 ....................... 4
5: 5 131 212 11111 ..................... 4
6: 6 141 33 222 1221 11211 111111 ...... 7
7: 7 151 313 11311 232 21112 1111111 ... 7
From _Reinhard Zumkeller_, Jan 23 2010: (Start)
Partitions into 1,2,4,6,... for the first values of n:
1: 1 ....................................... 1
2: 2 11 .................................... 2
3: 21 111 .................................. 2
4: 4 22 211 1111 ........................... 4
5: 41 221 2111 11111 ....................... 4
6: 6 42 4211 222 2211 21111 111111.......... 7
7: 61 421 42111 2221 22111 211111 1111111 .. 7. (End)

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 101 terms from Reinhard Zumkeller that were corrected by _Georg Fischer_, Jan 20 2019)

Cf. A172033, A004277. - Reinhard Zumkeller, Jan 23 2010
Cf. A004526, A030019, A056503, A147878, A320921, A322136.
The bisections are both A000070.
The ordered version (palindromic compositions) is A016116.
The complement is counted by A233771 and A210249.
The case of palindromic prime signature is A242414.
Palindromic partitions are ranked by A265640, with complement A229153.
The case of palindromic plane trees is A319436.
The multiplicative version (palindromic factorizations) is A344417.
A000569 counts graphical partitions.
A027187 counts partitions of even length, ranked by A028260.
A035363 counts partitions into even parts, ranked by A066207.
A058696 counts partitions of even numbers, ranked by A300061.
A110618 counts partitions with length <= half sum, ranked by A344291.
Cf. A000041, A067538, A143773, A209816, A338914, A338915, A340387, A344296, A344414, A344415, A344416.

a025065 = p (1:[2,4..]) where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011

import Data.List (group)
a025065 = length . filter (<= 1) .
                   map (sum . map ((`mod` 2) . length) . group) . ps 1
   where ps x 0 = [[]]
         ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 18 2013

Map[Length[Select[IntegerPartitions[#], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &]] &, Range[40]] (* Peter J. C. Moses, Jan 20 2014 *)
n = 8; Select[IntegerPartitions[n], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &] (* Peter J. C. Moses, Jan 20 2014 *)
CoefficientList[Series[1/((1 - x) Product[1 - x^(2 n), {n, 1, 50}]), {x, 0, 60}], x] (* Clark Kimberling, Mar 14 2014 *)

N=66; q='q+O('q^N); Vec( 1/((1-q)*eta(q^2)) ) \\ Joerg Arndt, Mar 11 2014

a(n) = A000070(A004526(n)). - Reinhard Zumkeller, Jan 23 2010
G.f.: 1/((1-q)*prod(n>=1, 1-q^(2*n))). [Joerg Arndt, Mar 11 2014]
a(2*k+2) = a(2*k) + A000041(k + 1). - David A. Corneth, May 29 2021
a(n) ~ exp(Pi*sqrt(n/3)) / (2*Pi*sqrt(n)). - Vaclav Kotesovec, Nov 16 2021

Edited by N. J. A. Sloane, Dec 29 2007

Prepended a(0)=1, added more terms, Joerg Arndt, Mar 11 2014

A320924 Heinz numbers of multigraphical partitions.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 27, 30, 36, 40, 48, 49, 63, 64, 70, 75, 81, 84, 90, 100, 108, 112, 120, 121, 144, 147, 154, 160, 165, 169, 175, 189, 192, 196, 198, 210, 220, 225, 243, 250, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 343, 351, 352, 360

Offset: 1

Gus Wiseman, Oct 24 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is multigraphical if it comprises the multiset of vertex-degrees of some multigraph.
Also Heinz numbers of integer partitions of even numbers whose greatest part is less than or equal to half the sum of parts, i.e., numbers n whose sum of prime indices A056239(n) is even and at least twice the greatest prime index A061395(n). - Gus Wiseman, May 23 2021

			The sequence of all multigraphical partitions begins: (), (11), (22), (211), (1111), (33), (222), (321), (2211), (3111), (21111), (44), (422), (111111), (431), (332), (2222), (4211), (3221), (3311), (22211), (41111), (32111), (55), (221111).
From _Gus Wiseman_, May 23 2021: (Start)
The sequence of terms together with their prime indices and a multigraph realizing each begins:
    1:      () | {}
    4:    (11) | {{1,2}}
    9:    (22) | {{1,2},{1,2}}
   12:   (112) | {{1,3},{2,3}}
   16:  (1111) | {{1,2},{3,4}}
   25:    (33) | {{1,2},{1,2},{1,2}}
   27:   (222) | {{1,2},{1,3},{2,3}}
   30:   (123) | {{1,3},{2,3},{2,3}}
   36:  (1122) | {{1,2},{3,4},{3,4}}
   40:  (1113) | {{1,4},{2,4},{3,4}}
   48: (11112) | {{1,2},{3,5},{4,5}}
   49:    (44) | {{1,2},{1,2},{1,2},{1,2}}
   63:   (224) | {{1,3},{1,3},{2,3},{2,3}}
(End)

These partitions are counted by A209816.
The case with odd weights is A322109.
The conjugate case of equality is A340387.
The conjugate version with odd weights allowed is A344291.
The conjugate opposite version is A344292.
The opposite version with odd weights allowed is A344296.
The conjugate version is A344413.
The conjugate opposite version with odd weights allowed is A344414.
The case of equality is A344415.
The opposite version is A344416.
A000070 counts non-multigraphical partitions.
A025065 counts palindromic partitions.
A035363 counts partitions into even parts.
A056239 adds up prime indices, row sums of A112798.
A110618 counts partitions that are the vertex-degrees of some set multipartition with no singletons.
A334201 adds up all prime indices except the greatest.
Cf. A000041, A000569, A007717, A096373, A265640, A283877, A306005, A318361, A320459, A320911, A320922, A320923, A320925.

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
Select[Range[1000],prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]!={}&]

Members m of A300061 such that A061395(m) <= A056239(m)/2. - Gus Wiseman, May 23 2021

A344296 Numbers with at least as many prime factors (counted with multiplicity) as half their sum of prime indices.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 256, 264, 270, 280, 288, 300, 320, 324, 336, 352

Offset: 1

Gus Wiseman, May 16 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.

These are the Heinz numbers of certain partitions counted by A025065, but different from palindromic partitions, which have Heinz numbers A265640.

			The sequence of terms together with their prime indices begins:
      1: {}            30: {1,2,3}
      2: {1}           32: {1,1,1,1,1}
      3: {2}           36: {1,1,2,2}
      4: {1,1}         40: {1,1,1,3}
      6: {1,2}         48: {1,1,1,1,2}
      8: {1,1,1}       54: {1,2,2,2}
      9: {2,2}         56: {1,1,1,4}
     10: {1,3}         60: {1,1,2,3}
     12: {1,1,2}       64: {1,1,1,1,1,1}
     16: {1,1,1,1}     72: {1,1,1,2,2}
     18: {1,2,2}       80: {1,1,1,1,3}
     20: {1,1,3}       81: {2,2,2,2}
     24: {1,1,1,2}     84: {1,1,2,4}
     27: {2,2,2}       88: {1,1,1,5}
     28: {1,1,4}       90: {1,2,2,3}

The case with difference at least 1 is A322136.
The case of equality is A340387, counted by A000041 or A035363.
The opposite version is A344291, counted by A110618.
The conjugate version is A344414, with even-weight case A344416.
A025065 counts palindromic partitions, ranked by A265640.
A056239 adds up prime indices, row sums of A112798.
A300061 lists numbers whose sum of prime indices is even.
Cf. A001399, A002865, A025147, A027336, A036036, A067712, A244990, A261144, A325691, A344293, A344295.

Select[Range[100],PrimeOmega[#]>=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&]

A056239(a(n)) <= 2*A001222(a(n)).

a(n) = A322136(n)/4.

A344415 Numbers whose greatest prime index is half their sum of prime indices.

Original entry on oeis.org

4, 9, 12, 25, 30, 40, 49, 63, 70, 84, 112, 121, 154, 165, 169, 198, 220, 264, 273, 286, 289, 325, 351, 352, 361, 364, 390, 442, 468, 520, 529, 561, 595, 624, 646, 714, 741, 748, 765, 832, 841, 850, 874, 918, 931, 952, 961, 988, 1020, 1045, 1173, 1197, 1224

Offset: 1

Gus Wiseman, May 19 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:
       4: {1,1}           198: {1,2,2,5}
       9: {2,2}           220: {1,1,3,5}
      12: {1,1,2}         264: {1,1,1,2,5}
      25: {3,3}           273: {2,4,6}
      30: {1,2,3}         286: {1,5,6}
      40: {1,1,1,3}       289: {7,7}
      49: {4,4}           325: {3,3,6}
      63: {2,2,4}         351: {2,2,2,6}
      70: {1,3,4}         352: {1,1,1,1,1,5}
      84: {1,1,2,4}       361: {8,8}
     112: {1,1,1,1,4}     364: {1,1,4,6}
     121: {5,5}           390: {1,2,3,6}
     154: {1,4,5}         442: {1,6,7}
     165: {2,3,5}         468: {1,1,2,2,6}
     169: {6,6}           520: {1,1,1,3,6}

The partitions with these Heinz numbers are counted by A035363.
The conjugate version is A340387.
This sequence is the case of equality in A344414 and A344416.
A001222 counts prime factors with multiplicity.
A025065 counts palindromic partitions, ranked by A265640.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A322109 ranks partitions of n with no part > n/2, counted by A110618.
A334201 adds up all prime indices except the greatest.
A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.
A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.
Cf. A000070, A001414, A209816, A301988, A316413, A316428, A320924, A325037, A325038, A325044, A330950, A344293, A344294, A344297.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max[primeMS[#]]==Total[primeMS[#]]/2&]

A061395(a(n)) = A056239(a(n))/2.

A344414 Heinz numbers of integer partitions whose sum is at most twice their greatest part.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Gus Wiseman, May 19 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 sequence of terms together with their prime indices begins:
     2: {1}        20: {1,1,3}    39: {2,6}
     3: {2}        21: {2,4}      40: {1,1,1,3}
     4: {1,1}      22: {1,5}      41: {13}
     5: {3}        23: {9}        42: {1,2,4}
     6: {1,2}      25: {3,3}      43: {14}
     7: {4}        26: {1,6}      44: {1,1,5}
     9: {2,2}      28: {1,1,4}    46: {1,9}
    10: {1,3}      29: {10}       47: {15}
    11: {5}        30: {1,2,3}    49: {4,4}
    12: {1,1,2}    31: {11}       51: {2,7}
    13: {6}        33: {2,5}      52: {1,1,6}
    14: {1,4}      34: {1,7}      53: {16}
    15: {2,3}      35: {3,4}      55: {3,5}
    17: {7}        37: {12}       56: {1,1,1,4}
    19: {8}        38: {1,8}      57: {2,8}
For example, 56 has prime indices {1,1,1,4} and 7 <= 2*4, so 56 is in the sequence. On the other hand, 224 has prime indices {1,1,1,1,1,4} and 9 > 2*4, so 224 is not in the sequence.

These partitions are counted by A025065 but are different from palindromic partitions, which have Heinz numbers A265640.
The opposite even-weight version appears to be A320924, counted by A209816.
The opposite version appears to be A322109, counted by A110618.
The case of equality in the conjugate version is A340387.
The conjugate opposite version is A344291, counted by A110618.
The conjugate opposite 5-smooth case is A344293, counted by A266755.
The conjugate version is A344296, also counted by A025065.
The case of equality is A344415.
The even-weight case is A344416.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A330950, A344294, A344297.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max[primeMS[#]]>=Total[primeMS[#]]/2&]

A056239(a(n)) <= 2*A061395(a(n)).

A322109 Heinz numbers of integer partitions that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons.

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 18, 24, 25, 27, 30, 32, 36, 40, 45, 48, 49, 50, 54, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 121, 125, 126, 128, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 169, 175, 180, 189, 192, 196, 198, 200, 210

Offset: 1

Gus Wiseman, Nov 26 2018

nonn

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

Also Heinz numbers of partitions whose greatest part is less than or equal to half the sum of parts, i.e., numbers n whose sum of prime indices A056239(n) is at least twice the greatest prime index A061395(n). - Gus Wiseman, May 23 2021

			Each term paired with its Heinz partition and a realizing set multipartition with no singletons:
   1:      (): {}
   4:    (11): {{1,2}}
   8:   (111): {{1,2,3}}
   9:    (22): {{1,2},{1,2}}
  12:   (211): {{1,2},{1,3}}
  16:  (1111): {{1,2,3,4}}
  18:   (221): {{1,2},{1,2,3}}
  24:  (2111): {{1,2},{1,3,4}}
  25:    (33): {{1,2},{1,2},{1,2}}
  27:   (222): {{1,2,3},{1,2,3}}
  30:   (321): {{1,2},{1,2},{1,3}}
  32: (11111): {{1,2,3,4,5}}
  36:  (2211): {{1,2},{1,2,3,4}}
  40:  (3111): {{1,2},{1,3},{1,4}}

These partitions are counted by A110618.
The even-weight version is A320924.
The conjugate case of equality is A340387.
The conjugate version is A344291.
The opposite conjugate version is A344296.
The opposite version is A344414.
The case of equality is A344415.
The opposite even-weight version is A344416.
A000070 counts non-multigraphical partitions.
A025065 counts palindromic partitions.
A035363 counts partitions into even parts.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A334201 adds up all prime indices except the greatest.
Cf. A000041, A000569, A265640, A283877, A306005, A318361, A320922, A320923, A320925.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqnopfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqnopfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimeQ[#]&&SquareFreeQ[#]&]}]]
Select[Range[100],Length[sqnopfacs[Times@@Prime/@nrmptn[#]]]>0&]

A061395(a(n)) <= A056239(a(n))/2.

A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).

Original entry on oeis.org

1, 3, 7, 9, 10, 13, 19, 21, 22, 25, 27, 28, 29, 30, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138

Offset: 1

Gus Wiseman, May 19 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.

Also Heinz numbers of integer partitions of even numbers m with at most m/2 parts, counted by A209816 riffled with zeros, or A110618 with odd positions zeroed out.

			The sequence of terms together with their prime indices begins:
      1: {}          37: {12}        75: {2,3,3}
      3: {2}         39: {2,6}       76: {1,1,8}
      7: {4}         43: {14}        79: {22}
      9: {2,2}       46: {1,9}       81: {2,2,2,2}
     10: {1,3}       49: {4,4}       82: {1,13}
     13: {6}         52: {1,1,6}     84: {1,1,2,4}
     19: {8}         53: {16}        85: {3,7}
     21: {2,4}       55: {3,5}       87: {2,10}
     22: {1,5}       57: {2,8}       88: {1,1,1,5}
     25: {3,3}       61: {18}        89: {24}
     27: {2,2,2}     62: {1,11}      90: {1,2,2,3}
     28: {1,1,4}     63: {2,2,4}     91: {4,6}
     29: {10}        66: {1,2,5}     94: {1,15}
     30: {1,2,3}     70: {1,3,4}    100: {1,1,3,3}
     34: {1,7}       71: {20}       101: {26}
For example, 75 has 3 prime indices {2,3,3} with sum 8 >= 2*3, so 75 is in the sequence.

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

These are the Heinz numbers of partitions counted by A209816 and A110618.
A subset of A300061 (sum of prime indices is even).
The conjugate version appears to be A320924 (allowing odd weights: A322109).
The case of equality is A340387.
Allowing odd weights gives A344291.
The 5-smooth case is A344295, or A344293 allowing odd weights.
The opposite version allowing odd weights is A344296.
The conjugate opposite version allowing odd weights is A344414.
The case of equality in the conjugate case is A344415.
The conjugate opposite version is A344416, counted by A000070.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A330950 counts partitions of n with Heinz number divisible by n.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A025065, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.

filter:= proc(n) local F,a,t;
  F:= ifactors(n)[2];
  a:= add((numtheory:-pi(t[1])-2)*t[2],t=F);
  a::even and a >= 0
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 10 2024

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

Members m of A300061 such that A056239(m) >= 2*A001222(m).

A366319 Numbers k such that the sum of prime indices of k is not twice the maximum prime index of k, meaning A056239(k) != 2 * A061395(k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Oct 10 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.

Also Heinz numbers of integer partitions containing n/2, where n is the sum of all parts.

			The prime indices of 90 are {1,2,2,3}, with sum 8 and twice maximum 6, so 90 is in the sequence.

Partitions of this type are counted by A086543.
For length instead of maximum we have the complement of A340387.
The complement is A344415, counted by A035363.
A001221 counts distinct prime factors, A001222 with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A334201 adds up all prime indices except the greatest.
A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.
A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.
Cf. A001414, A100959, A238628, A300061, A301987, A316413, A320924, A344414, A344416, A366318.

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

A344292 Numbers m whose sum of prime indices A056239(m) is even and is at most twice the number of prime factors counted with multiplicity A001222(m).

Original entry on oeis.org

1, 3, 4, 9, 10, 12, 16, 27, 28, 30, 36, 40, 48, 64, 81, 84, 88, 90, 100, 108, 112, 120, 144, 160, 192, 208, 243, 252, 256, 264, 270, 280, 300, 324, 336, 352, 360, 400, 432, 448, 480, 544, 576, 624, 640, 729, 756, 768, 784, 792, 810, 832, 840, 880, 900, 972

Offset: 1

Gus Wiseman, May 22 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.

Also Heinz numbers of integer partitions of even numbers m with at least m/2 parts, counted by A000070 riffled with 0's, or A025065 with odd positions zeroed out.

			The sequence of terms together with their prime indices begins:
       1: {}                 84: {1,1,2,4}
       3: {2}                88: {1,1,1,5}
       4: {1,1}              90: {1,2,2,3}
       9: {2,2}             100: {1,1,3,3}
      10: {1,3}             108: {1,1,2,2,2}
      12: {1,1,2}           112: {1,1,1,1,4}
      16: {1,1,1,1}         120: {1,1,1,2,3}
      27: {2,2,2}           144: {1,1,1,1,2,2}
      28: {1,1,4}           160: {1,1,1,1,1,3}
      30: {1,2,3}           192: {1,1,1,1,1,1,2}
      36: {1,1,2,2}         208: {1,1,1,1,6}
      40: {1,1,1,3}         243: {2,2,2,2,2}
      48: {1,1,1,1,2}       252: {1,1,2,2,4}
      64: {1,1,1,1,1,1}     256: {1,1,1,1,1,1,1,1}
      81: {2,2,2,2}         264: {1,1,1,2,5}

These are the Heinz numbers of partitions counted by A000070 and A025065.
A subset of A300061 (sum of prime indices is even).
The conjugate opposite version is A320924, counted by A209816.
The conjugate opposite version allowing odds is A322109, counted by A110618.
The case of equality is A340387, counted by A000041.
The opposite version allowing odd weights is A344291, counted by A110618.
Allowing odd weights gives A344296, counted by A025065.
The opposite version is A344413, counted by A209816.
The conjugate version allowing odd weights is A344414, counted by A025065.
The case of equality in the conjugate case is A344415, counted by A035363.
The conjugate version is A344416, counted by A000070.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A330950 counts partitions of n with Heinz number divisible by n.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A067538, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.

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

Members m of A300061 such that A056239(m) <= 2*A001222(m).

A366318 Heinz numbers of integer partitions that are of length 2 or begin with n/2, where n is the sum of all parts.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 21, 22, 25, 26, 30, 33, 34, 35, 38, 39, 40, 46, 49, 51, 55, 57, 58, 62, 63, 65, 69, 70, 74, 77, 82, 84, 85, 86, 87, 91, 93, 94, 95, 106, 111, 112, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 154, 155, 158, 159

Offset: 1

Gus Wiseman, Oct 08 2023

nonn

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

			The terms together with their prime indices begin:
     4: {1,1}      38: {1,8}         77: {4,5}
     6: {1,2}      39: {2,6}         82: {1,13}
     9: {2,2}      40: {1,1,1,3}     84: {1,1,2,4}
    10: {1,3}      46: {1,9}         85: {3,7}
    12: {1,1,2}    49: {4,4}         86: {1,14}
    14: {1,4}      51: {2,7}         87: {2,10}
    15: {2,3}      55: {3,5}         91: {4,6}
    21: {2,4}      57: {2,8}         93: {2,11}
    22: {1,5}      58: {1,10}        94: {1,15}
    25: {3,3}      62: {1,11}        95: {3,8}
    26: {1,6}      63: {2,2,4}      106: {1,16}
    30: {1,2,3}    65: {3,6}        111: {2,12}
    33: {2,5}      69: {2,9}        112: {1,1,1,1,4}
    34: {1,7}      70: {1,3,4}      115: {3,9}
    35: {3,4}      74: {1,12}       118: {1,17}

The first condition alone is A001358, counted by A004526.
The complement of the first condition is A100959, counted by A058984.
The partitions with these Heinz numbers are counted by A238628.
The second condition alone is A344415, counted by A035363.
The complement of the second condition is A366319, counted by A086543.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A322109 ranks partitions of n with no part > n/2, counted by A110618.
A334201 adds up all prime indices except the greatest.
A344296 solves for k in A001222(k) >= A056239(k)/2, counted by A025065.
Cf. A001414, A316413, A320924, A325044, A340387, A344414, A344416.

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

Union of A001358 and A344415.

cp's OEIS Frontend

A025065 Number of palindromic partitions of n.

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A320924 Heinz numbers of multigraphical partitions.

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A344296 Numbers with at least as many prime factors (counted with multiplicity) as half their sum of prime indices.

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A344415 Numbers whose greatest prime index is half their sum of prime indices.

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A344414 Heinz numbers of integer partitions whose sum is at most twice their greatest part.

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A322109 Heinz numbers of integer partitions that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons.

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A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).

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A366319 Numbers k such that the sum of prime indices of k is not twice the maximum prime index of k, meaning A056239(k) != 2 * A061395(k).

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