A344415 -id:A344415 - cp's OEIS frontend

A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 4, 6, 5, 5, 1, 7, 5, 8, 5, 6, 6, 9, 4, 3, 7, 2, 6, 10, 6, 11, 1, 7, 8, 7, 6, 12, 9, 8, 5, 13, 7, 14, 7, 7, 10, 15, 4, 4, 7, 9, 8, 16, 5, 8, 6, 10, 11, 17, 7, 18, 12, 8, 1, 9, 8, 19, 9, 11, 8, 20, 7, 21, 13, 8, 10, 9, 9, 22, 5

Offset: 1

Gus Wiseman, Jul 06 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

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 prime indices of 24 are {1,1,1,2}, with permutations such as (1,1,2,1) whose run-compression sums to 4, so a(24) = 4.
The prime indices of 216 are {1,1,1,2,2,2}, with permutations such as (1,2,1,2,1,2) whose run-compression sums to 9, so a(216) = 9.

Positions of first appearances are 1 followed by the primes A000040.
Positions of 1 are A000079 (powers of two) except 1.
Positions of 2 are A000244 (powers of three) except 1.
Positions of 3 are {6} U A000351 (six or powers of five) except 1.
For number of runs instead of sum of run-compression we have A373957.
For prime factors instead of indices we have A374250.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors (or prime indices).
A056239 adds up prime indices, row sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 lists run-compression of prime indices, sum A066328.
A335433 lists numbers whose prime indices are separable, complement A335448.
A373949 counts compositions by sum of run-compression, opposite A373951.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
Cf. A001223, A001414, A037201, A116608, A124767, A316413, A333755, A340387, A344415, A373954, A374246, A374247.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]

a(n) = A056239(n) iff n belongs to A335433 (the separable case), complement A335448.

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).

A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 6, 12, 14, 26, 31, 51, 61, 95, 114, 169, 202, 289, 347, 481, 576, 782, 936, 1244, 1487, 1946, 2323, 2997, 3570, 4551, 5414, 6827, 8103, 10127, 11997, 14866, 17575, 21619, 25507, 31166, 36692, 44563, 52362, 63240, 74152, 89112, 104281, 124731

Offset: 0

Gus Wiseman, Sep 19 2023

nonn

Also the number of integer partitions of n with no two possibly equal parts summing to n.

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (111)  (1111)  (221)    (222)     (322)      (332)
                           (311)    (411)     (331)      (521)
                           (2111)   (2211)    (421)      (611)
                           (11111)  (21111)   (511)      (2222)
                                    (111111)  (2221)     (3221)
                                              (3211)     (3311)
                                              (4111)     (5111)
                                              (22111)    (22211)
                                              (31111)    (32111)
                                              (211111)   (221111)
                                              (1111111)  (311111)
                                                         (2111111)
                                                         (11111111)

First condition alone is A058984, complement A004526, ranks A100959.
Second condition alone is A086543, complement A035363, ranks !A344415.
The complement is counted by A238628.
The strict case is A365826, complement A365659.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A140106 counts strict partitions of length 2, complement A365827.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A005408, A008967, A068911, A365377, A365544, A365543.

Table[Length[Select[IntegerPartitions[n],Length[#]!=2&&FreeQ[#,n/2]&]],{n,0,15}]

from sympy import npartitions
def A365825(n): return npartitions(n)-(m:=n>>1)-(0 if n&1 else npartitions(m)-1) # Chai Wah Wu, Sep 23 2023

Heinz numbers are A100959 /\ !A344415.

a(n) = A000041(n)-(n-1)/2 if n is odd. a(n) = A000041(n)-n/2-A000041(n/2)+1 if n is even. - Chai Wah Wu, Sep 23 2023

a(31)-a(47) from Chai Wah Wu, Sep 23 2023

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&]

A360669 Nonprime numbers > 1 for which the prime indices have the same mean as their first differences.

Original entry on oeis.org

10, 39, 68, 115, 138, 259, 310, 328, 387, 517, 574, 636, 793, 795, 1034, 1168, 1206, 1241, 1281, 1340, 1534, 1691, 1825, 2212, 2278, 2328, 2343, 2369, 2370, 2727, 2774, 2905, 3081, 3277, 3818, 3924, 4064, 4074, 4247, 4268, 4360, 4539, 4850, 4905, 5243, 5335

Offset: 1

Gus Wiseman, Feb 18 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.

			The terms together with their prime indices begin:
     1: {}
    10: {1,3}
    39: {2,6}
    68: {1,1,7}
   115: {3,9}
   138: {1,2,9}
   259: {4,12}
   310: {1,3,11}
   328: {1,1,1,13}
   387: {2,2,14}
   517: {5,15}
   574: {1,4,13}
   636: {1,1,2,16}
For example, the prime indices of 138 are {1,2,9}, with mean 4, and with first differences (1,7), with mean also 4, so 138 is in the sequence.

These partitions are counted by A360670.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, mean A088529/A088530.
A301987 lists numbers whose sum of prime indices equals their product.
A316413 lists numbers whose prime indices have integer mean.
A334201 adds up all prime indices except the greatest.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A340610, A344415, A348551, A359904, A360008, A360068, A360681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Mean[prix[#]]==Mean[Differences[prix[#]]]&]

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).

A363218 Positive integers whose prime indices satisfy: (length) = 2*(maximum).

Original entry on oeis.org

4, 24, 36, 54, 81, 160, 240, 360, 400, 540, 600, 810, 896, 900, 1000, 1215, 1344, 1350, 1500, 2016, 2025, 2240, 2250, 2500, 3024, 3136, 3360, 3375, 3750, 4536, 4704, 5040, 5600, 5625, 5632, 6250, 6804, 7056, 7560, 7840, 8400, 8448, 9375, 10206, 10584, 10976

Offset: 1

Gus Wiseman, May 23 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.

			The terms together with their prime indices begin:
      4: {1,1}
     24: {1,1,1,2}
     36: {1,1,2,2}
     54: {1,2,2,2}
     81: {2,2,2,2}
    160: {1,1,1,1,1,3}
    240: {1,1,1,1,2,3}
    360: {1,1,1,2,2,3}
    400: {1,1,1,1,3,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    810: {1,2,2,2,2,3}
    896: {1,1,1,1,1,1,1,4}
    900: {1,1,2,2,3,3}
   1000: {1,1,1,3,3,3}
   1215: {2,2,2,2,2,3}
   1344: {1,1,1,1,1,1,2,4}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   2016: {1,1,1,1,1,2,2,4}
   2025: {2,2,2,2,3,3}
   2240: {1,1,1,1,1,1,3,4}

The LHS (number of prime indices) is A001222.
The RHS is twice A061395.
Before multiplying by 2 we had A106529.
Partitions of this type are counted by A237753.
For sum instead of length we have A344415, counted by A035363.
An adjoint version is A361909, also counted by A237753.
For minimum instead of maximum we have A363134, counted by A237757.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
Cf. A001221, A067801, A118096, A324521, A237752, A361855, A361908, A363222.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[prix[#]]==2*Max[prix[#]]&]

Disjoint from A361909.

A365826 Number of strict integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 20, 20, 30, 31, 45, 46, 66, 68, 93, 97, 130, 136, 179, 188, 242, 256, 325, 344, 432, 459, 568, 606, 742, 793, 963, 1031, 1240, 1331, 1589, 1707, 2026, 2179, 2567, 2766, 3240, 3493, 4072, 4393, 5094, 5501, 6351

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

Also the number of strict integer partitions of n without two parts (allowing parts to be re-used) summing to n.

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

The second condition alone has bisections A078408 and A365828.
The complement is counted by A365659.
The non-strict version is A365825, complement A238628.
The first condition alone is A365827, complement A140106.
A000041 counts integer partitions, strict A000009.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A004526, A005408, A008967, A035363, A058984, A086543, A100959, A344415, A365376, A365377, A365543.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Tuples[#,2],n]&]], {n,0,30}]

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.

A363261 The partial sums of the prime indices of n include half the sum of all prime indices of n.

Original entry on oeis.org

4, 9, 12, 16, 25, 30, 40, 48, 49, 63, 64, 70, 81, 84, 108, 112, 121, 144, 154, 160, 165, 169, 192, 198, 220, 256, 264, 270, 273, 286, 289, 325, 351, 352, 360, 361, 364, 390, 442, 448, 468, 480, 520, 529, 561, 567, 576, 595, 624, 625, 640, 646, 675, 714, 729

Offset: 1

Gus Wiseman, Jun 06 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.

			The terms together with their prime indices begin:
   4: {1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  25: {3,3}
  30: {1,2,3}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
  63: {2,2,4}
  64: {1,1,1,1,1,1}
  70: {1,3,4}
  81: {2,2,2,2}
  84: {1,1,2,4}

Partitions of this type are counted by A322439.
For parts instead of partial sums we have A344415, counted by A035363.
A025065 counts palindromic partitions, ranked by A265640.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A301987 lists numbers whose sum of prime indices equals their product.
A322109 ranks partitions of n with no part > n/2, counted by A110618.
Cf. A001414, A106529, A327473, A327476, A334201, A340387, A360005.

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

cp's OEIS Frontend

A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

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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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A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

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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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A360669 Nonprime numbers > 1 for which the prime indices have the same mean as their first differences.

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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).

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A363218 Positive integers whose prime indices satisfy: (length) = 2*(maximum).

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A365826 Number of strict integer partitions of n that are not of length 2 and do not contain n/2.

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