A361204 -id:A361204 - cp's OEIS frontend

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

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

A237363 Number of partitions of n for which 2*(number of distinct parts) <= (number of parts).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 6, 6, 10, 13, 20, 26, 39, 50, 71, 87, 121, 156, 208, 265, 348, 440, 566, 712, 906, 1131, 1424, 1766, 2224, 2738, 3390, 4168, 5130, 6266, 7664, 9312, 11332, 13723, 16603, 20004, 24112, 28942, 34708, 41522, 49612, 59031, 70308, 83479, 98992

Offset: 0

Clark Kimberling, Feb 06 2014

nonn

a(n) + A237365(n) = A000041(n).

Also the number of integer partitions of n whose median difference is 0. For example, the partition (2,2,2,1,1) is counted because its multiset of differences {0,0,0,1} has median 0. - Gus Wiseman, Mar 18 2023

			Among the 22 partitions of 8, these qualify:  [5,1,1,1], [4,4], [4,1,1,1,1], [3,3,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1], and the remaining 12 do not, so that a(8) = 10.

Alois P. Heinz, Table of n, a(n) for n = 0..800

These partitions have ranks A361204.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts, reverse A058398.
A116608 counts partitions by number of distinct parts.
A359893 and A359901 count partitions by median, odd-length A359902.
Comparing twice the number of distinct parts to the number of parts:
              less: A360254, ranks A360558
             equal: A239959, ranks A067801
           greater: A237365, ranks A361393
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A000975, A027193, A114638, A240219, A325347, A360005, A360244.

z = 50; t = Map[Length[Select[IntegerPartitions[#], 2*Length[DeleteDuplicates[#]] <= Length[#] &]] &, Range[z]] (*A237363*)
Table[PartitionsP[n] - t[[n]], {n, 1, z}] (*A237365*) (* Peter J. C. Moses, Feb 06 2014 *)
Table[Length[Select[IntegerPartitions[n],Median[Differences[#]]==0&]],{n,0,30}] (* Gus Wiseman, Mar 18 2023 *)

A361205 a(n) = 2*omega(n) - bigomega(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, -1, 0, 2, 1, 1, 1, 2, 2, -2, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, -1, 1, 1, 3, 1, -3, 2, 2, 2, 0, 1, 2, 2, 0, 1, 3, 1, 1, 1, 2, 1, -1, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 2, 1, 2, 1, -4, 2, 3, 1, 1, 2, 3, 1, -1, 1, 2, 1, 1, 2, 3, 1, -1, -2, 2, 1, 2

Offset: 1

Gus Wiseman, Mar 16 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Without doubling omega we have -A046660.
Positions of 0's are A067801, counted by A239959.
Positions of negative terms are A360558, counted by A360254.
Positions of nonpositive terms are A361204, counted by A237363.
Positions of positive terms are A361393, counted by A237365.
Positions of nonnegative terms are A361395, counted by A361394.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
Cf. A061395, A067340, A111907, A324517, A324522, A326567/A326568.
Cf. A077761, A083342, A136141.

Table[2*PrimeNu[n]-PrimeOmega[n],{n,100}]

Additive with a(p^e) = 2 - e. - Amiram Eldar, Mar 26 2023

Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = 2*A077761 - A083342 = A077761 - A136141 = -0.511659... . - Amiram Eldar, Oct 01 2023

A361395 Positive integers k such that 2*omega(k) >= bigomega(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74

Offset: 1

Gus Wiseman, Mar 16 2023

nonn

Differs from A068938 in having 1 and 4 and lacking 80.

Includes all squarefree numbers.

			The prime indices of 80 are {1,1,1,1,3}, with 5 parts and 2 distinct parts, and 2*2 < 5, so 80 is not in the sequence.

Complement of A360558.
Positions of nonnegative terms in A361205.
These partitions are counted by A361394.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
              less: A360254, ranks A360558
             equal: A239959, ranks A067801
           greater: A237365, ranks A361393
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324515, A324517, A324521, A324560, A324562.
Cf. A360248, A360249, A360454.

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

A001222(a(n)) <= 2*A001221(a(n)).

A361393 Positive integers k such that 2*omega(k) > bigomega(k).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Gus Wiseman, Mar 16 2023

nonn

First differs from A317090 in having 120 and lacking 360.

There are numbers like 1, 120, 168, 180, 252,... which are not in A179983 but in here, and others like 360, 504, 540, 600,... which are in A179983 but not in here. - R. J. Mathar, Mar 21 2023

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
The prime indices of 120 are {1,1,1,2,3}, with 3 distinct parts and 5 parts, and 2*3 > 5, so 120 is in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, with 3 distinct parts and 6 parts, and 2*3 is not greater than 6, so 360 is not in the sequence.

These partitions are counted by A237365.
The complement is A361204.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
              less: A360254, ranks A360558
             equal: A239959, ranks A067801
           greater: A237365, ranks A361393
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

isA361393 := proc(n)
    if 2*A001221(n) > numtheory[bigomega](n) then
        true;
    else
        false ;
    end if:
end proc:
for n from 1 to 100 do
    if isA361393(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 21 2023

Select[Range[1000],2*PrimeNu[#]>PrimeOmega[#]&]

{k: 2*A001221(k) > A001222(k)}. - R. J. Mathar, Mar 21 2023

A385574 Number of integer partitions of n with the same number of adjacent equal parts as adjacent unequal parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 10, 11, 13, 17, 20, 30, 36, 44, 55, 70, 86, 98, 128, 156, 190, 235, 288, 351, 409, 499, 603, 722, 863, 1025, 1227, 1461, 1757, 2061, 2444, 2892, 3406, 3996, 4708, 5497, 6430, 7595, 8835, 10294, 12027, 13971, 16252, 18887, 21878

Offset: 0

Gus Wiseman, Jul 04 2025

nonn

These are also integer partitions of n with the same number of distinct parts as maximal anti-runs of parts.

			The partition (5,3,2,1,1,1,1) has 4 runs ((5),(3),(2),(1,1,1,1)) and 4 anti-runs ((5,3,2,1),(1),(1),(1)) so is counted under a(14).
The a(1) = 1 through a(10) = 10 reversed partitions (A = 10):
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (A)
                 (112)  (113)  (114)  (115)  (116)    (117)    (118)
                        (122)         (133)  (224)    (144)    (226)
                                      (223)  (233)    (225)    (244)
                                             (11123)  (11124)  (334)
                                                      (11223)  (11125)
                                                               (11134)
                                                               (11224)
                                                               (11233)
                                                               (12223)

Christian Sievers, Table of n, a(n) for n = 0..1000

The RHS is counted by A116608, rank statistic A297155.
The LHS is counted by A133121, rank statistic A046660.
For related inequalities see A212165, A212168, A361204.
For subsets instead of partitions see A217615, A385572, A385575.
These partitions are ranked by A385576.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A034839 counts subsets by number maximal runs, for partitions A384881, strict A116674.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A268193 counts partitions by maximal anti-runs, strict A384905, subsets A384893.
A355394 counts partitions with neighbors, complement A356236.
Cf. A000071, A003114, A008284, A010027, A047966, A210034, A325324, A325325, A356606, A384882, A384885.

Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union[#]]==Length[Split[#,#2!=#1&]]&]],{n,0,30}]

lista(n)=Vec(polcoef((prod(i=1,n,1+x^i/(t*(1-t*x^i))+O(x*x^n))-1)*t+1,0,t)) \\ Christian Sievers, Jul 18 2025

For a partition p, let s(p) be its sum, e(p) the number of equal adjacent pairs, and d(p) the number of distinct adjacent pairs. Then Sum_{p partition} x^s(p) * t^(e(p)-d(p)) = (Product_{i>=1} (1 + x^i/(t*(1-t*x^i))) - 1) * t + 1, so a(n) is the coefficient of x^n*t^0 of this expression. - Christian Sievers, Jul 18 2025

cp's OEIS Frontend

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

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A237363 Number of partitions of n for which 2*(number of distinct parts) <= (number of parts).

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A361205 a(n) = 2*omega(n) - bigomega(n).

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A361395 Positive integers k such that 2*omega(k) >= bigomega(k).

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A361393 Positive integers k such that 2*omega(k) > bigomega(k).

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A385574 Number of integer partitions of n with the same number of adjacent equal parts as adjacent unequal parts.

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