A088530 -id:A088530 - cp's OEIS frontend

A359889 Numbers that are 1 or whose prime indices have the same mean as median.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94

Offset: 1

Gus Wiseman, Jan 22 2023

nonn

First differs from A236510 in having 252 (prime indices {1,1,2,2,4}).
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 900 are {1,1,2,2,3,3}, with mean 2 and median 2, so 900 is in the sequence.

These partitions are counted by A240219, strict A359897.
The LHS (mean of prime indices) is A326567/A326568.
The complement is A359890, counted by A359894.
The odd-length case is A359891, complement A359892, counted by A359895.
The RHS (median of prime indices) is A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A359893 and A359901 count partitions by median, odd-length A359902.
A359908 lists numbers whose prime indices have integer median.
Cf. A026424, A327473, A359899, A359903, A359905, A359909, A360006, A360007, A360008, A360009.

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

Numbers n such that A326567(n)/A326568(n) = A360005(n)/2.

A088529 Numerator of Bigomega(n)/Omega(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 5, 2, 3, 1, 3, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 3, 6, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 3, 3, 1, 1, 1, 5, 4, 1, 1, 4, 1, 1, 1, 2, 1, 4, 1, 3, 1, 1, 1, 3, 1, 3, 3, 2

Offset: 2

Cino Hilliard, Nov 16 2003

			bigomega(24) / omega(24) = 4/2 = 2, so a(24) = 2.

H. Z. Cao, On the average of exponents, Northeast. Math. J., Vol. 10 (1994), pp. 291-296.

Antti Karttunen, Table of n, a(n) for n = 2..10000
R. L. Duncan, On the factorization of integers, Proc. Amer. Math. Soc. 25 (1970), 191-192.
Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022.
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A001222, A070012, A070013, A070014, A088530 (gives the denominator).

Cf. A136141, A272531.

Table[Numerator[PrimeOmega[n]/PrimeNu[n]], {n, 2, 100}] (* Michael De Vlieger, Jul 12 2017 *)

for(x=2,100,y=bigomega(x)/omega(x);print1(numerator(y)","))

from sympy import primefactors, Integer
def bigomega(n): return 0 if n==1 else bigomega(Integer(n)/primefactors(n)[0]) + 1
def omega(n): return Integer(len(primefactors(n)))
def a(n): return (bigomega(n)/omega(n)).numerator
print([a(n) for n in range(2, 51)]) # Indranil Ghosh, Jul 13 2017

Let B = number of prime divisors of n with multiplicity, O = number of distinct prime divisors of n. Then a(n) = numerator of B/O.
a(n) = A136565(n) = A181591(n) for n: 2 <= n < 24. - Reinhard Zumkeller, Nov 01 2010
Sum_{k=2..n} a(k)/A088530(k) ~ n + O(n/log(log(n))) (Duncan, 1970). - Amiram Eldar, Oct 14 2022
Sum_{k=2..n} a(k)/A088530(k) = n + c_1 * n/log(log(n)) + c_2 * n/log(log(n))^2 + O(n/log(log(n))^3), where c_1 = A136141 and c_2 = A272531 (Cao, 1994; Finch, 2020). - Amiram Eldar, Dec 15 2022

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}]

A359890 Numbers whose prime indices do not have the same mean as median.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 92, 96, 98, 99, 102, 104, 108, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154, 156, 160, 162, 164, 165

Offset: 1

Gus Wiseman, Jan 22 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 terms together with their prime indices begin:
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
For example, the prime indices of 360 are {1,1,1,2,2,3}, with mean 5/3 and median 3/2, so 360 is in the sequence.

The LHS (mean of prime indices) is A326567/A326568.
The complement is A359889, counted by A240219.
The odd-length case is A359891, complement A359892.
These partitions are counted by A359894.
The strict case is counted by A359898, odd-length A359900.
The RHS (median of prime indices) is A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A359908 lists numbers whose prime indices have integer median.
Cf. A327473, A327476, A348551, A359903, A359911, A359912, A360006-A360009.

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

A363943 Mean of the multiset of prime indices of n, rounded down.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 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.

Extending the terminology introduced at A124943, this is the "low mean" of prime indices.

			The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 1.

Positions of first appearances are 1 and A000040.
Before rounding down we had A326567/A326568.
For mode instead of mean we have A363486, high A363487.
For low median instead of mean we have A363941, triangle A124943.
For high median instead of mean we have A363942, triangle A124944.
The high version is A363944, triangle A363946.
The triangle for this statistic (low mean) is A363945.
Positions of 1's are A363949(n) = 2*A344296(n), counted by A025065.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363947 ranks partitions with rounded mean 1, counted by A363948.
Cf. A102627, A327473, A327476, A327482, A348551, A359889, A363727, A363951.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
meandown[y_]:=If[Length[y]==0,0,Floor[Mean[y]]];
Table[meandown[prix[n]],{n,100}]

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[#]]&]

A360460 Two times the median of the unordered prime signature of n; a(1) = 1.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 6, 4, 2, 2, 3, 2, 2, 2, 8, 2, 3, 2, 3, 2, 2, 2, 4, 4, 2, 6, 3, 2, 2, 2, 10, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 5, 4, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 2, 2, 2, 3, 12, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 5, 8, 2, 2, 2, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 14 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

			The unordered prime signature of 2520 is {1,1,2,3}, with median 3/2, so a(2520) = 3.

The version for divisors is A063655.
For mean instead of two times median we have A088529/A088530.
Prime signature is A124010, unordered A118914.
The version for prime indices is A360005.
The version for distinct prime indices is A360457.
The version for distinct prime factors is A360458.
The version for prime factors is A360459.
Positions of even terms are A360553.
Positions of odd terms are A360554.
The version for 0-prepended differences is A360555.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices.
A325347 counts partitions w/ integer median, complement A307683.
A329976 counts partitions with median multiplicity 1.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A133464, A359907, A359908, A360454.

Table[If[n==1,1,2*Median[Last/@FactorInteger[n]]],{n,100}]

A359905 Numbers whose prime indices and prime signature both have integer mean.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 71, 73, 78, 79, 81, 82, 83, 85, 87, 88, 89, 91, 94, 97, 100, 101, 103, 105, 107, 109, 110, 111, 113, 115, 118, 121

Offset: 1

Gus Wiseman, Jan 25 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:
     2: {1}          19: {8}            39: {2,6}
     3: {2}          21: {2,4}          41: {13}
     4: {1,1}        22: {1,5}          43: {14}
     5: {3}          23: {9}            46: {1,9}
     7: {4}          25: {3,3}          47: {15}
     8: {1,1,1}      27: {2,2,2}        49: {4,4}
     9: {2,2}        29: {10}           53: {16}
    10: {1,3}        30: {1,2,3}        55: {3,5}
    11: {5}          31: {11}           57: {2,8}
    13: {6}          32: {1,1,1,1,1}    59: {17}
    16: {1,1,1,1}    34: {1,7}          61: {18}
    17: {7}          37: {12}           62: {1,11}

A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 lists prime signature, mean A088529/A088530.
A316413 lists numbers whose prime indices have integer mean.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A327473, A348551, A359904, A359908, A360005, A360068, 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[100],IntegerQ[Mean[prix[#]]]&&IntegerQ[Mean[prisig[#]]]&]

Intersection of A316413 and A067340.

A360069 Number of integer partitions of n whose multiset of multiplicities has integer mean.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 9, 9, 13, 16, 25, 26, 39, 42, 62, 67, 95, 107, 147, 168, 225, 245, 327, 381, 471, 565, 703, 823, 1038, 1208, 1443, 1743, 2088, 2439, 2937, 3476, 4163, 4921, 5799, 6825, 8109, 9527, 11143, 13122, 15402, 17887, 20995, 24506, 28546, 33234, 38661

Offset: 0

Gus Wiseman, Jan 27 2023

nonn

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (2211)    (4111)     (521)
                                     (3111)    (211111)   (2222)
                                     (111111)  (1111111)  (3311)
                                                          (5111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
For example,  the partition (3,2,1,1,1,1) has multiplicities (1,1,4) with mean 2, so is counted under a(9). On the other hand, the partition (3,2,2,1,1) has multiplicities (1,2,2) with mean 5/3, so is not counted under a(9).

These partitions are ranked by A067340 (prime signature has integer mean).
Parts instead of multiplicities: A067538, strict A102627, ranked by A316413.
The case where the parts have integer mean also is ranked by A359905.
A000041 counts integer partitions, strict A000009.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature (A124010).
A326622 counts factorizations with integer mean, strict A328966.
Cf. A082550, A240219, A316313, A325347, A326669, A327475, A349156, A360068.

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

A360246 Numbers for which the prime indices do not have the same mean as the distinct prime indices.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189

Offset: 1

Gus Wiseman, Feb 07 2023

nonn

First differs from A242416 in having 126.
Contains no squarefree numbers or perfect powers.
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:
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   68: {1,1,7}
   72: {1,1,1,2,2}
The prime indices of 126 are {1,2,2,4} with mean 9/4 and distinct prime indices {1,2,4} with mean 7/3, so 126 is in the sequence.

Signature instead of parts: complement A324570, counted by A114638.
Signature instead of distinct parts: complement A359903, counted by A360068.
These partitions are counted by A360242.
The complement is A360247, counted by A360243.
For median we have A360248, counted by A360244 (complement A360245).
Union of A360252 and A360253, counted by A360250 and A360251.
A058398 counts partitions by mean, also A327482.
A088529/A088530 gives mean of prime signature (A124010).
A112798 lists prime indices, length A001222, sum A056239.
A316413 = numbers whose prime indices have integer mean, distinct A326621.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
Cf. A000975, A051293, A067340, A067538, A360005, A360241.

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

cp's OEIS Frontend

A359889 Numbers that are 1 or whose prime indices have the same mean as median.

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A088529 Numerator of Bigomega(n)/Omega(n).

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

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A359890 Numbers whose prime indices do not have the same mean as median.

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A363943 Mean of the multiset of prime indices of n, rounded down.

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

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A360460 Two times the median of the unordered prime signature of n; a(1) = 1.

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A359905 Numbers whose prime indices and prime signature both have integer mean.

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A360069 Number of integer partitions of n whose multiset of multiplicities has integer mean.