A067340 -id:A067340 - cp's OEIS frontend

A353862 Greatest run-sum of the prime indices of n.

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

Offset: 1

Gus Wiseman, May 23 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.

A run-sum of a sequence is the sum of any maximal consecutive constant subsequence.

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

Positions of first appearances are A008578.
For binary expansion we have A038374, least A144790.
For run-lengths instead of run-sums we have A051903.
Distinct run-sums are counted by A353835, weak A353861.
The least run-sum is given by A353931.
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.
A300273 ranks collapsible partitions, counted by A275870.
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, A182857, A316413, A353836, A353839, A353848, A353852, A353866.

Table[Max@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{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[#]]]&]

A360247 Numbers for which the prime indices have the same mean as the distinct prime indices.

Original entry on oeis.org

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, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 121, 122, 123, 125, 127, 128, 129, 130

Offset: 1

Gus Wiseman, Feb 07 2023

nonn

First differs from A072774 in having 90.
First differs from A242414 in lacking 126.
Includes all squarefree numbers and 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 prime indices of 900 are {3,3,2,2,1,1} with mean 2, and the distinct prime indices are {1,2,3} also with mean 2, so 900 is in the sequence.

Signature instead of parts: A324570, counted by A114638.
Signature instead of distinct parts: A359903, counted by A360068.
These partitions are counted by A360243.
The complement is A360246, counted by A360242.
For median instead of mean the complement is A360248, counted by A360244.
For median instead of mean we have A360249, counted by A360245.
For greater instead of equal mean we have A360252, counted by A360250.
For lesser instead of equal mean we have A360253, counted by A360251.
A008284 counts partitions by number of parts, distinct A116608.
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.

isA360247 := proc(n)
    local ifs,pidx,pe,meanAll,meanDist ;
    if n = 1 then
        return true ;
    end if ;
    ifs := ifactors(n)[2] ;
    # list of prime indices with multiplicity
    pidx := [] ;
    for pe in ifs do
        [numtheory[pi](op(1,pe)),op(2,pe)] ;
        pidx := [op(pidx),%] ;
    end do:
    meanAll := add(op(1,pe)*op(2,pe),pe=pidx) / add(op(2,pe),pe=pidx) ;
    meanDist := add(op(1,pe),pe=pidx) / nops(pidx) ;
    if meanAll = meanDist then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 130 do
    if isA360247(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 22 2023

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

A359913 Numbers whose multiset of prime factors has integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Jan 25 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).

			The terms together with their prime factors begin:
   2: {2}
   3: {3}
   4: {2,2}
   5: {5}
   7: {7}
   8: {2,2,2}
   9: {3,3}
  11: {11}
  12: {2,2,3}
  13: {13}
  15: {3,5}
  16: {2,2,2,2}
  17: {17}
  18: {2,3,3}
  19: {19}
  20: {2,2,5}
  21: {3,7}
  23: {23}
  24: {2,2,2,3}

Prime factors are listed by A027746.
The complement is A072978, for prime indices A359912.
For mean instead of median we have A078175, for prime indices A316413.
For prime indices instead of factors we have A359908, counted by A325347.
Positions of even terms in A360005.
A067340 lists numbers whose prime signature has integer mean.
A112798 lists prime indices, length A001222, sum A056239.
A325347 counts partitions with integer median, strict A359907.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A067538, A175352, A348551, A359890, A359905, A360007, A360006, A360069.

Select[Range[2,100],IntegerQ[Median[Flatten[ConstantArray@@@FactorInteger[#]]]]&]

A360553 Numbers > 1 whose unordered prime signature has integer median.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A067340 in having 60.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
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 unordered prime signature of 60 is {1,1,2}, with median 1, so 60 is in the sequence.
The unordered prime signature of 1260 is {1,1,2,2}, with median 3/2, so 1260 is not in the sequence.

For mean instead of median we have A067340, complement A070011.
Positions of even terms in A360460.
The complement is A360554 (without 1).
These partitions are counted by A360687.
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
A112798 lists prime indices, length A001222, sum A056239.
A124010 lists prime signature.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360454 = numbers whose prime indices and signature have the same median.
Cf. A000975, A026424, A078174, A078175, A316413, A326621, A360453.

Select[Range[2,100],IntegerQ[Median[Last/@FactorInteger[#]]]&]

A360687 Number of integer partitions of n whose multiplicities have integer median.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 10, 16, 22, 34, 42, 65, 80, 115, 145, 195, 240, 324, 396, 519, 635, 814, 994, 1270, 1549, 1952, 2378, 2997, 3623, 4521, 5466, 6764, 8139, 10008, 12023, 14673, 17534, 21273, 25336, 30593, 36302, 43575, 51555, 61570, 72653, 86382, 101676

Offset: 1

Gus Wiseman, Feb 20 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).

			The a(1) = 1 through a(8) = 16 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)    (3211)     (521)
                                     (3111)    (4111)     (2222)
                                     (111111)  (211111)   (3221)
                                               (1111111)  (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is counted under a(8).

The case of an odd number of multiplicities is A090794.
For mean instead of median we have A360069, ranks A067340.
These partitions have ranks A360553.
The complement is counted by A360690, ranks A360554.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

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

A070011 Numbers n such that number of prime factors divided by the number of distinct prime factors is not an integer.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 48, 50, 52, 60, 63, 68, 72, 75, 76, 80, 84, 90, 92, 98, 99, 108, 112, 116, 117, 120, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 168, 171, 172, 175, 176, 180, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228, 234, 236, 242

Offset: 1

Rick L. Shepherd, Apr 11 2002

nonn

			45 is a term because 45 = 3^2 * 5 gives bigomega(45)=3 and omega(45)=2 and 3/2 is not an integer.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A067340 (complement), A070012 (floor(bigomega(n)/omega(n))).

Different from A084679.

Select[Range[2,1000],!IntegerQ[PrimeOmega[#]/PrimeNu[#]]&] (* Enrique Pérez Herrero, Dec 20 2012 *)

v=[]; for(n=2,300,if(denominator(bigomega(n)/omega(n))<>1,v=concat(v,n))); v

is(n)=my(f=factor(n)[,2]); n>9 && vecsum(f)%#f!=0 \\ Charles R Greathouse IV, Oct 16 2015

A001222(n)/A001221(n) (i.e. bigomega(n)/omega(n)) is not an integer.

A070012 Floor of number of prime factors of n divided by the number of n's distinct prime factors.

Original entry on oeis.org

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

Offset: 2

Rick L. Shepherd, Apr 11 2002

nonn

a(n) is the integer part of the average of the exponents in the prime factorization of n.

			a(12)=1 because 12=2^2 * 3^1 and floor(bigomega(12)/omega(12)) = floor((2+1)/2) = 1.
a(36)=2 because 36=2^2 * 3^2 and floor(bigomega(36)/omega(36)) = floor((2+2)/2) = 2.
a(60)=1 because 60=2^2 * 3^1 * 5^1 and floor(bigomega(60)/omega(60)) = floor((2+1+1)/3) = 1.
36 is in A067340. 12 and 60 are in A070011.

G. C. Greubel, Table of n, a(n) for n = 2..5000

Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is an integer before floor applied), A070011 (ratio is not an integer), A070013 (ratio rounded), A070014 (ceiling of ratio), A046660 (bigomega(n)-omega(n)).

A070012[n_]:=Floor[PrimeOmega[n]/PrimeNu[n]];Array[A070012,100]

v=[]; for(n=2,150,v=concat(v,(bigomega(n)\omega(n)))); v

a(n) = floor(bigomega(n)/omega(n)) for n>=2.

cp's OEIS Frontend

A353862 Greatest run-sum of the prime indices of n.

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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.

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

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A360247 Numbers for which the prime indices have the same mean as the distinct prime indices.

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A359913 Numbers whose multiset of prime factors has integer median.

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A360553 Numbers > 1 whose unordered prime signature has integer median.

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A360687 Number of integer partitions of n whose multiplicities have integer median.

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A070011 Numbers n such that number of prime factors divided by the number of distinct prime factors is not an integer.

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