A363949 -id:A363949 - cp's OEIS frontend

A126594 Floor of the average of the prime factors of n with multiplicity.

2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 4, 4, 2, 17, 2, 19, 3, 5, 6, 23, 2, 5, 7, 3, 3, 29, 3, 31, 2, 7, 9, 6, 2, 37, 10, 8, 2, 41, 4, 43, 5, 3, 12, 47, 2, 7, 4, 10, 5, 53, 2, 8, 3, 11, 15, 59, 3, 61, 16, 4, 2, 9, 5, 67, 7, 13, 4, 71, 2, 73, 19, 4, 7, 9, 6, 79, 2, 3, 21, 83, 3, 11, 22, 16, 4, 89, 3, 10

Offset: 2

Cino Hilliard, Jan 06 2007

Cf. A067629 (rounding instead of flooring), A076690.
This is the floor of A123528/A123529.
Without multiplicity we have A363895.
For prime indices instead of factors we have A363943, triangle A363945.
Positions of first appearances are A364037.
The ceiling is A364156.
Positions of 2's are A364157, for prime indices A363949.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, ranks A316413.
A078175 lists numbers with integer mean of prime factors.
Cf. A026905, A326567/A326568, A327473, A327476, A363944, A363948, A363950.

Table[Floor[(Plus@@Times@@@FactorInteger[n])/PrimeOmega[n]], {n, 2, 90}] (* Alonso del Arte, May 21 2012 *)

avg(n) = { local(x,j,ln) for(x=2,n,a=ifactor(x); ln=length(a); print1(floor(sum(j=1,ln,a[j])/ln)",")) } ifactor(n) = \The vector of the prime factors of n with multiplicity. { local(f,j,k,flist); flist=[]; f=Vec(factor(n)); for(j=1,length(f[1]), for(k = 1,f[2][j],flist = concat(flist,f[1][j]) ); ); return(flist) }

a(p^n)=p, p prime, n >= 1. - Philippe Deléham, Nov 23 2008

a(n) = floor(A001414(n)/A001222(n)). - Philippe Deléham, Nov 24 2008

A364056 Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.

Original entry on oeis.org

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 64, 68, 72, 76, 80, 88, 92, 96, 104, 112, 116, 120, 124, 128, 136, 144, 148, 152, 160, 164, 168, 172, 176, 184, 188, 192, 200, 208, 212, 224, 232, 236, 240, 244, 248, 256, 264, 268, 272, 280, 284, 288, 292

Offset: 1

Gus Wiseman, Jul 07 2023

nonn

The multiset of prime factors of n is row n of A027746.

The high median (see A124944) in a multiset is either the middle part (for odd length), or the greatest of the two middle parts (for even length).

			The terms together with their prime indices begin:
     2: {1}             64: {1,1,1,1,1,1}      136: {1,1,1,7}
     4: {1,1}           68: {1,1,7}            144: {1,1,1,1,2,2}
     8: {1,1,1}         72: {1,1,1,2,2}        148: {1,1,12}
    12: {1,1,2}         76: {1,1,8}            152: {1,1,1,8}
    16: {1,1,1,1}       80: {1,1,1,1,3}        160: {1,1,1,1,1,3}
    20: {1,1,3}         88: {1,1,1,5}          164: {1,1,13}
    24: {1,1,1,2}       92: {1,1,9}            168: {1,1,1,2,4}
    28: {1,1,4}         96: {1,1,1,1,1,2}      172: {1,1,14}
    32: {1,1,1,1,1}    104: {1,1,1,6}          176: {1,1,1,1,5}
    40: {1,1,1,3}      112: {1,1,1,1,4}        184: {1,1,1,9}
    44: {1,1,5}        116: {1,1,10}           188: {1,1,15}
    48: {1,1,1,1,2}    120: {1,1,1,2,3}        192: {1,1,1,1,1,1,2}
    52: {1,1,6}        124: {1,1,11}           200: {1,1,1,3,3}
    56: {1,1,1,4}      128: {1,1,1,1,1,1,1}    208: {1,1,1,1,6}

Partitions of this type are counted by A027336.
Median of prime indices is A360005(n)/2.
For mode instead of median we have A360013, low A360015.
The low version is A363488, positions of 1's in A363941.
Positions of 1's in A363942.
A112798 lists prime indices, length A001222, sum A056239.
A123528/A123529 gives mean of prime factors, indices A326567/A326568.
A124943 counts partitions by low median, high A124944.
Cf. A072978, A215366, A316413, A359908, A363727, A363740, A363949.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
merr[y_]:=If[Length[y]==0,0,If[OddQ[Length[y]],y[[(Length[y]+1)/2]], y[[1+Length[y]/2]]]];
Select[Range[100],merr[prifacs[#]]==2&]

A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 14, 19, 27, 36, 51, 66, 90, 118, 156, 201, 264, 336, 434, 550, 700, 880, 1112, 1385, 1733, 2149, 2666, 3283, 4049, 4956, 6072, 7398, 9009, 10922, 13237, 15970, 19261, 23147, 27790, 33260, 39776, 47425, 56497, 67133, 79685, 94371, 111653

Offset: 1

Clark Kimberling, Feb 27 2014

Also the number of integer partitions of n + 1 with median > 1, or with no more 1's than non-1 parts. - Gus Wiseman, Jul 10 2023

			a(6) = 9 counts all the 11 partitions of 6 except 42 and 411.
From _Gus Wiseman_, Jul 10 2023 (Start)
The a(2) = 1 through a(8) = 14 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)
       (21)  (22)  (32)   (33)    (43)    (44)
             (31)  (41)   (42)    (52)    (53)
                   (221)  (51)    (61)    (62)
                          (222)   (322)   (71)
                          (321)   (331)   (332)
                          (2211)  (421)   (422)
                                  (2221)  (431)
                                  (3211)  (521)
                                          (2222)
                                          (3221)
                                          (3311)
                                          (4211)
                                          (22211)
(End)

Cf. A096373.
For mean instead of median we have A000065, ranks A057716.
The complement is counted by A027336, ranks A364056.
Rows sums of A359893 if we remove the first column.
These partitions have ranks A364058.
A000041 counts integer partitions.
A008284 counts partitions by length, A058398 by mean.
A025065 counts partitions with low mean 1, ranks A363949.
A124943 counts partitions by low median, high A124944.
A241131 counts partitions with low mode 1, ranks A360015.
Cf. A000070, A000975, A002865, A110618, A237984, A363488.

Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Length[p] + Min[p]]], {n, 50}]
Table[Length[Select[IntegerPartitions[n+1],Median[#]>1&]],{n,30}] (* Gus Wiseman, Jul 10 2023 *)

From Gus Wiseman, Jul 11 2023: (Start)
a(n>2) = A000041(n) - A096373(n-2).
a(n>1) = A000041(n-2) + A002865(n+1).
a(n) = A000041(n+1) - A027336(n).
(End)

Formula corrected by Gus Wiseman, Jul 11 2023

A363954 Numbers whose prime indices have low mean 2.

Original entry on oeis.org

3, 9, 10, 14, 15, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 70, 75, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 189, 196, 198, 204, 208, 210, 220, 225, 234, 243, 250, 252, 260, 264, 270, 272, 280, 294, 297, 300, 304, 308, 312, 315, 330, 350

Offset: 1

Gus Wiseman, Jul 05 2023

nonn

Extending the terminology of A124944, the "low mean" of a multiset is obtained by taking the mean and rounding down.

			The terms together with their prime indices begin:
     3: {2}
     9: {2,2}
    10: {1,3}
    14: {1,4}
    15: {2,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    42: {1,2,4}
    44: {1,1,5}
    45: {2,2,3}
    50: {1,3,3}
    52: {1,1,6}
    63: {2,2,4}
    66: {1,2,5}
    70: {1,3,4}
    75: {2,3,3}
    81: {2,2,2,2}
    84: {1,1,2,4}
    88: {1,1,1,5}
    90: {1,2,2,3}
   100: {1,1,3,3}

Partitions of this type are counted by A363745.
Positions of 2's in A363943 (high A363944), triangle A363945 (high A363946).
For mean 1 we have A363949.
The high version is A363950, counted by A026905.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A326567/A326568 gives mean of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
A363948 lists numbers whose prime indices have mean 1, counted by A363947.
Cf. A327473, A327476, A359889, A360013, A360015, A363488, A363951.

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

A363489 Rounded mean of the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 07 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.

We use the "rounding half to even" rule, see link.

			The prime indices of 180 are {1,1,2,2,3}, with mean 9/5, which rounds to 2, so a(180) = 2.

Wikipedia, Rounding.

Positions of first appearances are 1 and A000040.
Before rounding we had A326567/A326568.
For rounded-down: A363943, triangle A363945.
For rounded-up: A363944, triangle A363946.
Positions of 1's are A363948, complement A364059.
The triangle for this statistic (rounded mean) is A364060.
For prime factors instead of indices we have A364061.
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.
Cf. A327473, A327476, A327482, A359889, A360005, A363947, A363949, A363951.

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

A364059 Number of integer partitions of n whose rounded mean is > 1. Partitions with mean >= 3/2.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 9, 11, 18, 26, 35, 49, 70, 89, 123, 164, 212, 278, 366, 460, 597, 762, 957, 1210, 1530, 1891, 2369, 2943, 3621, 4468, 5507, 6703, 8210, 10004, 12115, 14688, 17782, 21365, 25743, 30913, 36965, 44210, 52801, 62753, 74667, 88626, 104874, 124070

Offset: 0

Gus Wiseman, Jul 06 2023

nonn

We use the "rounding half to even" rule, see link.

			The a(0) = 0 through a(8) = 18 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)     (7)     (8)
             (21)  (22)  (32)   (33)    (43)    (44)
                   (31)  (41)   (42)    (52)    (53)
                         (221)  (51)    (61)    (62)
                         (311)  (222)   (322)   (71)
                                (321)   (331)   (332)
                                (411)   (421)   (422)
                                (2211)  (511)   (431)
                                (3111)  (2221)  (521)
                                        (3211)  (611)
                                        (4111)  (2222)
                                                (3221)
                                                (3311)
                                                (4211)
                                                (5111)
                                                (22211)
                                                (32111)
                                                (41111)

Wikipedia, Rounding.

Rounding-up gives A000065.
Rounding-down gives A110618, ranks A344291.
For median instead of mean we appear to have A238495.
The complement is counted by A363947, ranks A363948.
A000041 counts integer partitions.
A008284 counts partitions by length, A058398 by mean.
A025065 counts partitions with low mean 1, ranks A363949.
A067538 counts partitions with integer mean, ranks A316413.
A124943 counts partitions by low median, high A124944.
Cf. A002865, A098859, A241131, A327482, A363723, A363724, A363731, A363946.

Table[Length[Select[IntegerPartitions[n],Round[Mean[#]]>1&]],{n,0,30}]

a(n) = A000041(n) - A363947(n).

A364055 Number of integer partitions of n satisfying (length) = (mean). Partitions of n into sqrt(n) parts.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 34, 0, 0, 0, 0, 0, 0, 0, 0, 192, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1206, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8033, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55974, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jul 07 2023

nonn

			The a(0) = 1 through a(9) = 7 partitions:
  ()  (1)  .  .  (22)  .  .  .  .  (333)
                 (31)              (432)
                                   (441)
                                   (522)
                                   (531)
                                   (621)
                                   (711)

The strict case is A107379(sqrt(n)).
Without  zeros we have A206240.
These partitions have ranks A363951.
A008284 counts partitions by length, A058398 by mean.
A067538 counts partitions with integer mean, ranks A316413.
Cf. A025065, A026905, A237984, A327472, A327482, A363944, A363949.

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

a(n^2) = A206240(n).

A364157 Numbers whose rounded-down (floor) mean of prime factors (with multiplicity) is 2.

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 40, 48, 54, 64, 72, 80, 96, 108, 120, 128, 144, 160, 162, 192, 216, 224, 240, 256, 288, 320, 324, 360, 384, 432, 448, 480, 486, 512, 576, 640, 648, 672, 720, 768, 800, 864, 896, 960, 972, 1024, 1080, 1152, 1280, 1296, 1344

Offset: 1

Gus Wiseman, Jul 18 2023

nonn

			The terms together with their prime factors begin:
   2 = 2
   4 = 2*2
   6 = 2*3
   8 = 2*2*2
  12 = 2*2*3
  16 = 2*2*2*2
  18 = 2*3*3
  24 = 2*2*2*3
  32 = 2*2*2*2*2
  36 = 2*2*3*3
  40 = 2*2*2*5
  48 = 2*2*2*2*3
  54 = 2*3*3*3
  64 = 2*2*2*2*2*2
  72 = 2*2*2*3*3
  80 = 2*2*2*2*5
  96 = 2*2*2*2*2*3

Without multiplicity we appear to have A007694.
Prime factors are listed by A027746, indices A112798.
Positions of 2's in A126594, positions of first appearances A364037.
For prime indices and ceiling we have A363950, counted by A026905.
For prime indices we have A363954 (or A363949), counted by A363745.
A078175 lists numbers with integer mean of prime factors.
A123528/A123529 gives mean of prime factors, indices A326567/A326568.
A316413 ranks partitions with integer mean, counted by A067538.
A363895 gives floor of mean of distinct prime factors.
A363943 gives floor of mean of prime indices, ceiling A363944.
Cf. A001222, A056239, A327473, A327476, A360013, A360015, A363488, A363945.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Floor[Mean[prifacs[#]]]==2&]

cp's OEIS Frontend

A126594 Floor of the average of the prime factors of n with multiplicity.

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A364056 Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.

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A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

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A363954 Numbers whose prime indices have low mean 2.

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A363489 Rounded mean of the multiset of prime indices of n.

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A364059 Number of integer partitions of n whose rounded mean is > 1. Partitions with mean >= 3/2.

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A364055 Number of integer partitions of n satisfying (length) = (mean). Partitions of n into sqrt(n) parts.

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A364157 Numbers whose rounded-down (floor) mean of prime factors (with multiplicity) is 2.

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