A359889 -id:A359889 - cp's OEIS frontend

A363741 Number of factorizations of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

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

Offset: 1

Gus Wiseman, Jun 26 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
Position of first appearance of n is: (1, 2, 4, 16, 64, 5832, 4096, ...).

			The factorization 6*9*9*12 = 5832 has mean 9, median 9, and modes {9}, so it is counted under a(5832).
The a(n) factorizations for selected n:
2   4     16        64            5832              4096
    2*2   4*4       8*8           18*18*18          64*64
          2*2*2*2   4*4*4         6*9*9*12          8*8*8*8
                    2*2*2*2*2*2   3*6*6*6*9         16*16*16
                                  2*3*3*3*3*3*3*4   4*4*4*4*4*4
                                                    2*2*2*2*2*2*2*2*2*2*2*2

For just (mean) = (median): A359909, see A240219, A359889, A359910, A359911.
The version for partitions is A363719, unequal A363720.
For unequal instead of equal we have A363742.
A000041 counts integer partitions.
A001055 counts factorizations, strict A045778, ordered A074206.
A089723 counts constant factorizations.
A316439 counts factorizations by length, A008284 partitions.
A326622 counts factorizations with integer mean, strict A328966.
A339846 counts even-length factorizations, A339890 odd-length.
Cf. A237984, A326567/A326568, A327472, A359893, A363723, A363727, A363740.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[facs[n],{Mean[#]}=={Median[#]}==modes[#]&]],{n,100}]

A363951 Numbers whose prime indices satisfy (length) = (mean), or (sum) = (length)^2.

Original entry on oeis.org

2, 9, 10, 68, 78, 98, 99, 105, 110, 125, 328, 444, 558, 620, 783, 812, 870, 966, 988, 1012, 1035, 1150, 1156, 1168, 1197, 1254, 1326, 1330, 1425, 1521, 1666, 1683, 1690, 1704, 1785, 1870, 1911, 2002, 2125, 2145, 2275, 2401, 2412, 2541, 2662, 2680, 2695, 3025

Offset: 1

Gus Wiseman, Jul 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:
    2: {1}
    9: {2,2}
   10: {1,3}
   68: {1,1,7}
   78: {1,2,6}
   98: {1,4,4}
   99: {2,2,5}
  105: {2,3,4}
  110: {1,3,5}
  125: {3,3,3}
  328: {1,1,1,13}
  444: {1,1,2,12}
  558: {1,2,2,11}
  620: {1,1,3,11}
  783: {2,2,2,10}
  812: {1,1,4,10}
  870: {1,2,3,10}
  966: {1,2,4,9}
  988: {1,1,6,8}

Partitions of this type are counted by A364055, without  zeros A206240.
The RHS is A001222.
The LHS is A326567/A326568.
A008284 counts partitions by length, A058398 by mean.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, sum A056239.
A124943 counts partitions by low median, high A124944.
A316413 ranks partitions with integer mean, counted by A067538.
A326622 counts factorizations with integer mean, strict A328966.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A025065, A327473, A327476, A327482, A359889, A363723, A363727, A363729, A363944, A363949.

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

A359892 Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not have the same mean as median.

Original entry on oeis.org

12, 18, 20, 28, 42, 44, 45, 48, 50, 52, 63, 66, 68, 70, 72, 75, 76, 78, 80, 92, 98, 99, 102, 108, 112, 114, 116, 117, 120, 124, 130, 138, 147, 148, 153, 154, 162, 164, 165, 168, 170, 171, 172, 174, 175, 176, 180, 182, 186, 188, 190, 192, 195, 200, 207, 208

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}
   28: {1,1,4}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   70: {1,3,4}
   72: {1,1,1,2,2}
For example, the prime indices of 180 are {1,1,2,2,3}, with mean 9/5 and median 2, so 180 is in the sequence.

A subset of A026424 = numbers with odd bigomega.
The LHS (mean of prime indices) is A326567/A326568.
This is the odd-length case of A359890, complement A359889.
The complement is A359891.
These partitions are counted by A359896, complement A359895.
The RHS (median of prime indices) is A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A359902 counts odd-length partitions by median.
Cf. A240219, A327473, A327476, A348551, A359894, A359898, A359899, A359900, A359911, A359912, A360006-A360009.

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

Intersection of A026424 and A359890.

A359911 Number of integer factorizations of n into factors > 1 without the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 1, 0, 1, 1, 0, 0, 6, 0, 1, 0, 1, 0, 3, 0, 3, 0, 0, 0, 4, 0, 0, 1, 4, 0, 1, 0, 1, 0, 1, 0, 9, 0, 0, 1, 1, 0, 1, 0, 6, 1, 0, 0, 5, 0, 0, 0, 3, 0, 5, 0, 1, 0, 0, 0, 13, 0, 1, 1, 3, 0, 1, 0, 3, 0, 0, 0, 10

Offset: 1

Gus Wiseman, Jan 24 2023

nonn

			The a(72) = 9 factorizations: (2*2*2*3*3), (2*2*2*9), (2*2*3*6), (2*2*18), (2*3*12), (2*4*9), (2*6*6), (3*3*8), (3*4*6).

Antti Karttunen, Table of n, a(n) for n = 1..65537

The version for partitions is A359894, complement A240219.
The complement is counted by A359909, odd-length A359910.
A001055 counts factorizations.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A316313, A326567/A326568, A359897, A359889, A359906, A360005.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Mean[#]!=Median[#]&]],{n,100}]

median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);
A359911(n, m=n, facs=List([])) = if(1==n, (#facs>0 && (median(facs)!=(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A359911(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

For n > 1, a(n) = A001055(n) - A359909(n). - Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

A363722 Nonprime numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 90, 121, 125, 128, 169, 243, 256, 270, 289, 343, 361, 512, 525, 529, 550, 625, 729, 756, 810, 841, 961, 1024, 1331, 1369, 1666, 1681, 1849, 1911, 1950, 2048, 2187, 2197, 2209, 2268, 2401, 2430, 2625, 2695, 2700, 2750, 2809

Offset: 1

Gus Wiseman, Jun 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 mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
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:
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    16: {1,1,1,1}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    49: {4,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
    90: {1,2,2,3}
   121: {5,5}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}

These partitions are counted by A363719 - 1 for n > 0.
Including primes gives A363727, counted by A363719.
For prime powers instead of just primes we have A363729, counted by A363728.
For unequal instead of equal we have A363730, counted by A363720.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A000961, A327473, A327476, A359893, A359908, A360009, A360550, A363721, A363725, A363741.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[100],!PrimeQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

Complement of A000040 in A363727.

Assuming there is a unique mode, we have A326567(a(n))/A326568(a(n)) = A360005(a(n))/2 = A363486(a(n)) = A363487(a(n)).

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}]

cp's OEIS Frontend

A363741 Number of factorizations of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

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A363951 Numbers whose prime indices satisfy (length) = (mean), or (sum) = (length)^2.

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A359892 Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not have the same mean as median.

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A359911 Number of integer factorizations of n into factors > 1 without the same mean as median.

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A363722 Nonprime numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

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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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