A067801 -id:A067801 - 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 *)

A361856 Positive integers whose prime indices satisfy (maximum) = 2*(median).

Original entry on oeis.org

12, 24, 42, 48, 60, 63, 72, 96, 126, 130, 140, 144, 189, 192, 195, 252, 266, 288, 308, 325, 330, 360, 378, 384, 399, 420, 432, 495, 546, 567, 572, 576, 588, 600, 630, 638, 650, 665, 756, 768, 819, 864, 882, 884, 931, 945, 957, 962, 975, 1122, 1134, 1152, 1190

Offset: 1

Gus Wiseman, Apr 02 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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
These are Heinz numbers of partitions satisfying (maximum) = 2*(median).

			The terms together with their prime indices begin:
    12: {1,1,2}
    24: {1,1,1,2}
    42: {1,2,4}
    48: {1,1,1,1,2}
    60: {1,1,2,3}
    63: {2,2,4}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
   126: {1,2,2,4}
   130: {1,3,6}
   140: {1,1,3,4}
   144: {1,1,1,1,2,2}
The prime indices of 126 are {1,2,2,4}, with maximum 4 and median 2, so 126 is in the sequence.
The prime indices of 308 are {1,1,4,5}, with maximum 5 and median 5/2, so 308 is in the sequence.

The LHS (greatest prime index) is A061395.
The RHS (twice median) is A360005, distinct A360457.
These partitions are counted by A361849.
For mean instead of median we have A361855, counted by A361853.
For minimum instead of median we have A361908, counted by A118096.
For length instead of median we have A361909, counted by A237753.
A000975 counts subsets with integer median.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
Cf. A027193, A067801, A111907, A361205, A361848, A361858.

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

A061395(a(n)) = 2*A360005(a(n)).

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)).

A361855 Numbers > 1 whose prime indices satisfy (maximum) * (length) = 2*(sum).

Original entry on oeis.org

28, 40, 78, 84, 171, 190, 198, 220, 240, 252, 280, 351, 364, 390, 406, 435, 714, 748, 756, 765, 777, 784, 814, 840, 850, 925, 988, 1118, 1197, 1254, 1330, 1352, 1419, 1425, 1440, 1505, 1564, 1600, 1638, 1716, 1755, 1794, 1802, 1820, 1950, 2067, 2204, 2254

Offset: 1

Gus Wiseman, Mar 29 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 positive integers whose prime indices satisfy (maximum) = 2*(mean).
Also Heinz numbers of partitions of the same size as their complement (see example).

			The terms together with their prime indices begin:
   28: {1,1,4}
   40: {1,1,1,3}
   78: {1,2,6}
   84: {1,1,2,4}
  171: {2,2,8}
  190: {1,3,8}
  198: {1,2,2,5}
  220: {1,1,3,5}
  240: {1,1,1,1,2,3}
  252: {1,1,2,2,4}
  280: {1,1,1,3,4}
The prime indices of 84 are {1,1,2,4}, with maximum 4, length 4, and sum 8, and 4*4 = 2*8, so 84 is in the sequence.
The prime indices of 120 are {1,1,1,2,3}, with maximum 3, length 5, and sum 8, and 3*5 != 2*8, so 120 is not in the sequence.
The prime indices of 252 are {1,1,2,2,4}, with maximum 4, length 5, and sum 10, and 4*5 = 2*10, so 252 is in the sequence.
The partition (5,2,2,1) with Heinz number 198 has diagram:
  o o o o o
  o o . . .
  o o . . .
  o . . . .
Since the partition and its complement (shown in dots) both have size 10, 198 is in the sequence.

These partitions are counted by A361853, strict A361854.
For median instead of mean we have A361856, counted by A361849.
For minimum instead of mean we have A361908, counted by A118096.
For length instead of mean we have A361909, counted by A237753.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A061395 gives greatest prime index.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
Cf. A067801, A237824, A316413, A324521, A326844, A361205, A361851, A361906.

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

A061395(a(n)) * A001222(a(n)) = 2*A056239(a(n)).

A361908 Positive integers > 1 whose prime indices satisfy (maximum) = 2*(minimum).

Original entry on oeis.org

6, 12, 18, 21, 24, 36, 48, 54, 63, 65, 72, 96, 105, 108, 133, 144, 147, 162, 189, 192, 216, 288, 315, 319, 324, 325, 384, 432, 441, 455, 481, 486, 525, 567, 576, 648, 715, 731, 735, 768, 845, 864, 931, 945, 972, 1007, 1029, 1152, 1296, 1323, 1403, 1458, 1463

Offset: 1

Gus Wiseman, Apr 05 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:
     6: {1,2}
    12: {1,1,2}
    18: {1,2,2}
    21: {2,4}
    24: {1,1,1,2}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    63: {2,2,4}
    65: {3,6}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}

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

The RHS is 2*A055396 (twice minimum).
The LHS is A061395 (greatest prime index).
Partitions of this type are counted by A118096.
For mean instead of minimum we have A361855, counted by A361853.
For median instead of minimum we have A361856, counted by A361849.
For length instead of minimum we have A361909, counted by A237753.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors with multiplicity.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A237820, A237821, A361858.

filter:= proc(n) local F,b;
  if n::even then b:= padic:-ordp(n,3);
     if b = 0 then return false else return n = 2^padic:-ordp(n,2) * 3^b fi
  fi;
  F:= ifactors(n)[2][..,1];
  nops(F) >= 2 and numtheory:-pi(max(F)) = 2*numtheory:-pi(min(F))
end proc:
select(filter, [$1..2000]); # Robert Israel, Mar 11 2025

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]==2*PrimePi[FactorInteger[#][[1,1]]]&]

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

A361204 Positive integers k such that 2*omega(k) <= bigomega(k).

Original entry on oeis.org

1, 4, 8, 9, 16, 24, 25, 27, 32, 36, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 100, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304

Offset: 1

Gus Wiseman, Mar 14 2023

nonn

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    16: {1,1,1,1}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    48: {1,1,1,1,2}
    49: {4,4}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}

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

These partitions are counted by A237363.
The complement is A361393.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
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. A324517, A324521, A324522, A324560, A324562.

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(t[2],t=F) >= 2*nops(F)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 22 2023

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

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

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)).

A361867 Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

Original entry on oeis.org

20, 28, 40, 44, 52, 56, 66, 68, 76, 78, 80, 84, 88, 92, 99, 102, 104, 112, 114, 116, 117, 120, 124, 132, 136, 138, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 190, 198, 200, 204, 207, 208, 212, 220, 222, 224, 228, 230, 232, 234

Offset: 1

Gus Wiseman, Apr 05 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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 > 2*(3/2), so 84 is in the sequence.
The terms together with their prime indices begin:
   20: {1,1,3}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   52: {1,1,6}
   56: {1,1,1,4}
   66: {1,2,5}
   68: {1,1,7}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   92: {1,1,9}
   99: {2,2,5}

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal version is A361856, counted by A361849.
These partitions are counted by A361857, reverse A361858.
Including the equal case gives A361868, counted by A361859.
For mean instead of median we have A361907.
A000975 counts subsets with integer median.
A001222 counts prime factors, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A111907, A237751, A237820, A359893, A361848, A361855, A361908, A361909.

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

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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A361856 Positive integers whose prime indices satisfy (maximum) = 2*(median).

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

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A361855 Numbers > 1 whose prime indices satisfy (maximum) * (length) = 2*(sum).

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A361908 Positive integers > 1 whose prime indices satisfy (maximum) = 2*(minimum).

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

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A361204 Positive integers k such that 2*omega(k) <= bigomega(k).

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