A067340 -id:A067340 - cp's OEIS frontend

A323023 Irregular triangle read by rows where row n is the omega-sequence of n.

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

Offset: 1

Gus Wiseman, Jan 02 2019

We define the omega-sequence of n to have length A323014(n), and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of A181819.

Except for n = 1, all rows end with 1. If n is not prime, the term in row n prior to the last is A304465(n).

			The sequence of omega-sequences begins:
   1:            26: 2 2 1      51: 2 2 1        76: 3 2 2 1
   2: 1          27: 3 1        52: 3 2 2 1      77: 2 2 1
   3: 1          28: 3 2 2 1    53: 1            78: 3 3 1
   4: 2 1        29: 1          54: 4 2 2 1      79: 1
   5: 1          30: 3 3 1      55: 2 2 1        80: 5 2 2 1
   6: 2 2 1      31: 1          56: 4 2 2 1      81: 4 1
   7: 1          32: 5 1        57: 2 2 1        82: 2 2 1
   8: 3 1        33: 2 2 1      58: 2 2 1        83: 1
   9: 2 1        34: 2 2 1      59: 1            84: 4 3 2 2 1
  10: 2 2 1      35: 2 2 1      60: 4 3 2 2 1    85: 2 2 1
  11: 1          36: 4 2 1      61: 1            86: 2 2 1
  12: 3 2 2 1    37: 1          62: 2 2 1        87: 2 2 1
  13: 1          38: 2 2 1      63: 3 2 2 1      88: 4 2 2 1
  14: 2 2 1      39: 2 2 1      64: 6 1          89: 1
  15: 2 2 1      40: 4 2 2 1    65: 2 2 1        90: 4 3 2 2 1
  16: 4 1        41: 1          66: 3 3 1        91: 2 2 1
  17: 1          42: 3 3 1      67: 1            92: 3 2 2 1
  18: 3 2 2 1    43: 1          68: 3 2 2 1      93: 2 2 1
  19: 1          44: 3 2 2 1    69: 2 2 1        94: 2 2 1
  20: 3 2 2 1    45: 3 2 2 1    70: 3 3 1        95: 2 2 1
  21: 2 2 1      46: 2 2 1      71: 1            96: 6 2 2 1
  22: 2 2 1      47: 1          72: 5 2 2 1      97: 1
  23: 1          48: 5 2 2 1    73: 1            98: 3 2 2 1
  24: 4 2 2 1    49: 2 1        74: 2 2 1        99: 3 2 2 1
  25: 2 1        50: 3 2 2 1    75: 3 2 2 1     100: 4 2 1

Row lengths are A323014, or A182850 if we assume A182850(2) = 1.
First column is empty if n = 1 and otherwise A001222(n).
Second column is empty if n is 1 or prime and otherwise A001221(n).
Third column is empty if n is 1, prime, or a power of a prime and otherwise A071625(n).
Cf. A024619, A056239, A067340, A118914, A181819, A181821, A182857, A304464, A304465, A323022.

red[n_]:=Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]];
omg[n_,k_]:=If[k==1,PrimeOmega[n],omg[red[n],k-1]];
dep[n_]:=If[n==1,0,If[PrimeQ[n],1,1+dep[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]]];
Table[omg[n,k],{n,100},{k,dep[n]}]

A360068 Number of integer partitions of n such that the parts have the same mean as the multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 6, 0, 0, 0, 6, 0, 7, 0, 1, 0, 0, 0, 0, 90, 0, 63, 0, 0, 0, 0, 11, 0, 0, 0, 436, 0, 0, 0, 0, 0, 0, 0, 0, 2157, 0, 0, 240, 1595, 22, 0, 0, 0, 6464, 0, 0, 0, 0, 0, 0, 0, 0, 11628, 4361, 0, 0, 0, 0, 0, 0, 0, 12927, 0, 0, 621, 0

Offset: 0

Gus Wiseman, Jan 27 2023

nonn

Note that such a partition cannot be strict for n > 1.

Conjecture: If n is squarefree, then a(n) = 0.

			The n = 1, 4, 8, 9, 12, 16, 18 partitions (D=13):
  (1)  (22)  (3311)  (333)  (322221)  (4444)      (444222)
             (5111)         (332211)  (43222111)  (444411)
                            (422211)  (43321111)  (552222)
                            (522111)  (53221111)  (555111)
                            (531111)  (54211111)  (771111)
                            (621111)  (63211111)  (822222)
                                                  (D11111)
For example, the partition (4,3,3,3,3,3,2,2,1,1) has mean 5/2, and its multiplicities (1,5,2,2) also have mean 5/2, so it is counted under a(20).

These partitions are ranked by A359903, for prime factors A359904.
Positions of positive terms are A360070.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature (A124010).
A326567/A326568 gives mean of prime indices (A112798).
A360069 counts partitions whose multiplicities have integer mean.
Cf. A067340, A067538, A082550, A240219, A316313, A327475, A349156, A359893, A359897, A359905.

Table[Length[Select[IntegerPartitions[n],Mean[#]==Mean[Length/@Split[#]]&]],{n,0,30}]

A353866 Heinz numbers of rucksack partitions. Every prime-power divisor has a different sum of prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset: 1

Gus Wiseman, Jun 06 2022

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.

In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum, so these are run-knapsack partitions or rucksack partitions for short.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
The sequence contains 18 because its prime-power divisors {1,2,3,9} have prime indices {}, {1}, {2}, {2,2} with distinct sums {0,1,2,4}. On the other hand, 12 is not in the sequence because {2} and {1,1} have the same sum.

Knapsack partitions are counted by A108917, ranked by A299702.
The strong case is A353838, counted by A353837, complement A353839.
These partitions are counted by A353864.
The complete case is A353867, counted by A353865.
The complement is A354583.
A000041 counts partitions, strict A000009.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents the operation of taking run-sums of a partition.
A353836 counts partitions by number of distinct run-sums.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A018818, A067340, A181819, A304442, A316413, A325862, A353833, A353835, A353861, A353931.

msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Total/@Select[msubs[primeMS[#]],SameQ@@#&]&]

A111907 Numbers k such that the same number of primes, among primes <= the largest prime dividing k, divide k as do not.

Original entry on oeis.org

1, 3, 9, 14, 21, 27, 28, 35, 56, 63, 78, 81, 98, 112, 130, 147, 156, 175, 182, 189, 195, 196, 224, 234, 243, 245, 260, 273, 286, 312, 364, 392, 429, 441, 448, 455, 468, 520, 567, 570, 572, 585, 624, 650, 686, 702, 715, 728, 729, 784, 798, 819, 875, 896, 936

Offset: 1

Leroy Quet, Aug 19 2005

nonn

Also numbers whose greatest prime index (A061395) is twice their number of distinct prime factors (A001221). - Gus Wiseman, Mar 19 2023

			28 is included because 7 is the largest prime dividing 28. And of the primes <= 7 (2,3,5,7), 2 and 7 (2 primes) divide 28 and 3 and 5 (also 2 primes) do not divide 28.
From _Gus Wiseman_, Mar 19 2023: (Start)
The terms together with their prime indices begin:
    1: {}
    3: {2}
    9: {2,2}
   14: {1,4}
   21: {2,4}
   27: {2,2,2}
   28: {1,1,4}
   35: {3,4}
   56: {1,1,1,4}
   63: {2,2,4}
   78: {1,2,6}
   81: {2,2,2,2}
   98: {1,4,4}
  112: {1,1,1,1,4}
  130: {1,3,6}
  147: {2,4,4}
  156: {1,1,2,6}
For example, 156 is included because it has prime indices {1,1,2,6}, with distinct parts {1,2,6} and distinct non-parts {3,4,5}, both of length 3. Alternatively, 156 has greatest prime index 6 and omega 3, and 6 = 2*3.
(End)

John Tyler Rascoe, Table of n, a(n) for n = 1..1000

For length instead of maximum we have A067801.
These partitions are counted by A239959.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A061395 gives greatest prime index.
A112798 lists prime indices, sum A056239.
Comparing twice the number of distinct parts to greatest part:
              less: A360254, ranks A111906
             equal: A239959, ranks A111907
           greater: A237365, ranks A111905
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A046660, A067340, A324517, A324521, A324562, A361205, A361393.

Select[Range[100],2*PrimeNu[#]==PrimePi[FactorInteger[#][[-1,1]]]&] (* Gus Wiseman, Mar 19 2023 *)

{m=950;v=vector(m);for(n=1,m,f=factor(n)[,1]~;c=0;pc=0;forprime(p=2,vecmax(f), j=1;s=length(f);while(j<=s&&p!=f[j],j++);if(j<=s,c++);pc++);v[n]=sign(pc-2*c)); for(n=1,m,if(v[n]==0,print1(n,",")))} \\ Klaus Brockhaus, Aug 21 2005

from itertools import count, islice
from sympy import sieve, factorint
def a_gen():
    yield 1
    for k in count(3):
        f = [sieve.search(i)[0] for i in factorint(k)]
        if 2*len(f) == f[-1]:
            yield k
A111907_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jun 20 2024

More terms from Klaus Brockhaus, Aug 21 2005

A359903 Numbers whose prime indices and prime signature have the same mean.

Original entry on oeis.org

1, 2, 9, 88, 100, 125, 624, 756, 792, 810, 880, 900, 1312, 2401, 4617, 4624, 6240, 7392, 7560, 7920, 8400, 9261, 9604, 9801, 10648, 12416, 23424, 33984, 37760, 45792, 47488, 60912, 66176, 71552, 73920, 75200, 78720, 83592, 89216, 89984, 91264, 91648, 99456

Offset: 1

Gus Wiseman, Jan 24 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.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      9: {2,2}
     88: {1,1,1,5}
    100: {1,1,3,3}
    125: {3,3,3}
    624: {1,1,1,1,2,6}
    756: {1,1,2,2,2,4}
    792: {1,1,1,2,2,5}
    810: {1,2,2,2,2,3}
    880: {1,1,1,1,3,5}
    900: {1,1,2,2,3,3}
   1312: {1,1,1,1,1,13}
   2401: {4,4,4,4}
   4617: {2,2,2,2,2,8}
   4624: {1,1,1,1,7,7}
   6240: {1,1,1,1,1,2,3,6}
   7392: {1,1,1,1,1,2,4,5}
   7560: {1,1,1,2,2,2,3,4}
   7920: {1,1,1,1,2,2,3,5}
Example: 810 has prime indices {1,2,2,2,2,3} and prime exponents (1,4,1), both of which have mean 2, so 810 is in the sequence.
Example: 78720 has prime indices {1,1,1,1,1,1,1,2,3,13} and prime exponents (7,1,1,1), both of which have mean 5/2, so 78720 is in the sequence.

Prime indices are A112798, sum A056239, mean A326567/A326568.
Prime signature is A124010, sum A001222, mean A088529/A088530.
For prime factors instead of indices we have A359904.
Partitions with these Heinz numbers are counted by A360068.
A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A316413 lists numbers whose prime indices have integer mean.
A360005 gives median of prime indices (times two).
Cf. A240219, A316313, A326622, A327473, A327476, A348551, A359905, A359908, A360008, A360069.

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

A360241 Number of integer partitions of n whose distinct parts have integer mean.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 6, 13, 13, 22, 19, 43, 34, 56, 66, 97, 92, 156, 143, 233, 256, 322, 341, 555, 542, 710, 831, 1098, 1131, 1644, 1660, 2275, 2484, 3035, 3492, 4731, 4848, 6063, 6893, 8943, 9378, 12222, 13025, 16520, 18748, 22048, 24405, 31446, 33698, 41558

Offset: 0

Gus Wiseman, Feb 02 2023

nonn

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (331)      (44)
                    (31)    (11111)  (42)      (511)      (53)
                    (1111)           (51)      (3211)     (62)
                                     (222)     (31111)    (71)
                                     (321)     (1111111)  (422)
                                     (3111)               (2222)
                                     (111111)             (3221)
                                                          (3311)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)
For example, the partition (32111) has distinct parts {1,2,3} with mean 2, so is counted under a(8).

For parts instead of distinct parts we have A067538, ranked by A316413.
The strict case is A102627.
These partitions are ranked by A326621.
For multiplicities instead of distinct parts: A360069, ranked by A067340.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, also A327482.
A116608 counts partitions by number of distinct parts.
A326619/A326620 gives mean of distinct prime indices.
A326622 counts factorizations with integer mean, strict A328966.
A360071 counts partitions by number of parts and number of distinct parts.
The following count partitions:
- A360242 mean(parts) != mean(distinct parts), ranked by A360246.
- A360243 mean(parts) = mean(distinct parts), ranked by A360247.
- A360250 mean(parts) > mean(distinct parts), ranked by A360252.
- A360251 mean(parts) < mean(distinct parts), ranked by A360253.
Cf. A067539, A078174, A082550, A240219, A316313, A325347, A326567/A326568, A326669, A327475, A349156.

Table[Length[Select[IntegerPartitions[n],IntegerQ[Mean[Union[#]]]&]],{n,0,30}]

A353931 Least run-sum of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 07 2022

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.

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 3.

Positions of first appearances are A008578.
For run-lengths instead of run-sums we have A051904, greatest A051903.
For run-sums and binary expansion we have A144790, greatest A038374.
For run-lengths and binary expansion we have A175597, greatest A043276.
Distinct run-sums are counted by A353835, weak A353861.
The greatest run-sum is given by A353862.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A304442 counts partitions with all equal run-sums, compositions A353851.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run sums, nonprime A353834.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A067340, A073093, A181819, A300273, A316413, A353839, A353866.

Table[Min@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]

A360242 Number of integer partitions of n where the parts do not have the same mean as the distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 11, 19, 25, 43, 49, 82, 103, 136, 183, 258, 314, 435, 524, 687, 892, 1150, 1378, 1788, 2241, 2773, 3399, 4308, 5142, 6501, 7834, 9600, 11726, 14099, 16949, 20876, 25042, 30032, 35732, 43322, 51037, 61650, 72807, 86319, 102983, 122163

Offset: 0

Gus Wiseman, Feb 04 2023

nonn

			The a(1) = 0 through a(9) = 19 partitions:
  .  .  .  (211)  (221)   (411)    (322)     (332)      (441)
                  (311)   (3111)   (331)     (422)      (522)
                  (2111)  (21111)  (511)     (611)      (711)
                                   (2221)    (4211)     (3222)
                                   (3211)    (5111)     (3321)
                                   (4111)    (22211)    (4221)
                                   (22111)   (32111)    (4311)
                                   (31111)   (41111)    (5211)
                                   (211111)  (221111)   (6111)
                                             (311111)   (22221)
                                             (2111111)  (32211)
                                                        (33111)
                                                        (42111)
                                                        (51111)
                                                        (321111)
                                                        (411111)
                                                        (2211111)
                                                        (3111111)
                                                        (21111111)
For example, the partition y = (32211) has mean 9/5 and distinct parts {1,2,3} with mean 2, so y is counted under a(9).

The complement for multiplicities instead of distinct parts is A360068.
The complement is counted by A360243, ranks A360247.
For median instead of mean we have A360244, complement A360245.
These partitions have ranks A360246.
Sum of A360250 and A360251, ranks A360252 and A360253.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A058398 counts partitions by mean, also A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A360071 counts partitions by number of parts and number of distinct parts.
A360241 counts partitions whose distinct parts have integer mean.
Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156.

Table[Length[Select[IntegerPartitions[n],Mean[#]!=Mean[Union[#]]&]],{n,0,30}]

A360243 Number of integer partitions of n where the parts have the same mean as the distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 11, 11, 17, 13, 28, 19, 32, 40, 48, 39, 71, 55, 103, 105, 110, 105, 197, 170, 195, 237, 319, 257, 462, 341, 515, 543, 584, 784, 1028, 761, 973, 1153, 1606, 1261, 2137, 1611, 2368, 2815, 2575, 2591, 4393, 3798, 4602, 4663, 5777, 5121

Offset: 0

Gus Wiseman, Feb 04 2023

nonn

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (11111)  (51)      (61)       (62)
                                     (222)     (421)      (71)
                                     (321)     (1111111)  (431)
                                     (2211)               (521)
                                     (111111)             (2222)
                                                          (3221)
                                                          (3311)
                                                          (11111111)

For multiplicities instead of distinct parts we have A360068.
The complement is counted by A360242, ranks A360246.
For median instead of mean we have A360245, complement A360244.
These partitions have ranks A360247.
Cf. A360250 and A360251, ranks A360252 and A360253.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A058398 counts partitions by mean, also A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A360071 counts partitions by number of parts and number of distinct parts.
A360241 counts partitions whose distinct parts have integer mean.
Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156, A360069.

Table[Length[Select[IntegerPartitions[n],Mean[#]==Mean[Union[#]]&]],{n,0,30}]

A360558 Numbers whose multiset of prime factors (or indices, see A112798) has more adjacent equalities (or parts that have appeared before) than distinct parts.

Original entry on oeis.org

8, 16, 27, 32, 48, 64, 72, 80, 81, 96, 108, 112, 125, 128, 144, 160, 162, 176, 192, 200, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 343, 352, 368, 384, 392, 400, 405, 416, 432, 448, 464, 480, 486, 496, 500, 512, 544, 567, 576, 592, 608, 625, 640, 648

Offset: 1

Gus Wiseman, Feb 20 2023

nonn

No terms are squarefree.

Also numbers whose first differences of 0-prepended prime indices have median 0.

			The terms together with their prime indices begin:
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    32: {1,1,1,1,1}
    48: {1,1,1,1,2}
    64: {1,1,1,1,1,1}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}
    81: {2,2,2,2}
    96: {1,1,1,1,1,2}
   108: {1,1,2,2,2}
   112: {1,1,1,1,4}
   125: {3,3,3}
For example, the prime indices of 720 are {1,1,1,1,2,2,3} with 4 adjacent equalities and 3 distinct parts, so 720 is in the sequence.

For equality we have A067801.
These partitions are counted by A360254.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A360005 gives median of prime indices (times 2).
Cf. A000975, A027193, A067340, A237363, A317090, A360248, A360249, A360454, A360555, A360556.

Select[Range[100],PrimeOmega[#]>2*PrimeNu[#]&]

A001222(a(n)) > 2*A001221(a(n)).

cp's OEIS Frontend

A323023 Irregular triangle read by rows where row n is the omega-sequence of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360068 Number of integer partitions of n such that the parts have the same mean as the multiplicities.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A353866 Heinz numbers of rucksack partitions. Every prime-power divisor has a different sum of prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A111907 Numbers k such that the same number of primes, among primes <= the largest prime dividing k, divide k as do not.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A359903 Numbers whose prime indices and prime signature have the same mean.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360241 Number of integer partitions of n whose distinct parts have integer mean.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A353931 Least run-sum of the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360242 Number of integer partitions of n where the parts do not have the same mean as the distinct parts.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A360243 Number of integer partitions of n where the parts have the same mean as the distinct parts.

Views

Author

Keywords

Examples