A363727 -id:A363727 - cp's OEIS frontend

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

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[#]&]

A363721 Number of odd-length integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 2, 2, 5, 7, 1, 2, 8, 2, 9, 16, 11, 2, 2, 15, 16, 37, 33, 2, 44, 2, 1, 79, 33, 103, 127, 2, 47, 166, 39, 2, 214, 2, 384, 738, 90, 2, 2, 277, 185, 631, 1077, 2, 1065, 1560, 477, 1156, 223, 2, 2863

Offset: 1

Gus Wiseman, Jun 21 2023

nonn

The median of an odd-length partition is the middle part.

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(n) partitions for n = {1, 3, 9, 14, 15, 18, 20, 22} (A..M = 10..22):
  1  3    9          E        F                I          K      M
     111  333        2222222  555              666        44444  22222222222
          111111111  3222221  33333            222222222  54443  32222222221
                     3322211  43332            322222221  64442  33222222211
                     4222211  53331            332222211  65441  33322222111
                              63321            422222211  74432  42222222211
                              111111111111111  432222111  74441  43222222111
                                               522222111  84431  44222221111
                                                          94421  52222222111
                                                                 53222221111
                                                                 62222221111

All odd-length partitions are counted by A027193.
For just (mean) = (median) we have A359895, also A240219, A359899, A359910.
For just (mean) != (median) we have A359896, also A359894, A359900.
Allowing any length gives A363719, ranks A363727, non-constant A363728.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or negative mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
A363726 counts odd-length partitions with a unique mode.
Cf. A237984, A325347, A326567/A326568, A327472, A363720, A363740, A363741.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&{Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]

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)).

A363742 Number of integer factorizations of n with different mean, median, and mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 7, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0, 0, 7

Offset: 1

Gus Wiseman, Jun 27 2023

nonn

An integer factorization of n is a multiset of positive integers > 1 with product n.
If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
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, 30, 48, 60, 72, 200, 160, 96, ...

			The a(n) factorizations for n = 30, 48, 60, 72, 96, 144:
  (2*3*5)  (2*3*8)    (2*5*6)    (2*4*9)    (2*6*8)    (2*8*9)
           (2*2*3*4)  (2*3*10)   (3*4*6)    (3*4*8)    (3*6*8)
                      (2*2*3*5)  (2*3*12)   (2*3*16)   (2*3*24)
                                 (2*2*3*6)  (2*4*12)   (2*4*18)
                                            (2*2*3*8)  (2*6*12)
                                            (2*2*4*6)  (3*4*12)
                                            (2*3*4*4)  (2*2*4*9)
                                                       (2*3*4*6)
                                                       (2*2*3*12)
                                                       (2*2*3*3*4)

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

Just (mean) != (median): A359911, complement A359909, partitions A359894.
The version for partitions is A363720, equal A363719, ranks A363730.
For equal instead of unequal we have A363741.
A001055 counts factorizations, strict A045778, ordered A074206.
A316439 counts factorizations by length, A008284 partitions.
A363265 counts factorizations with a unique mode.
Cf. A089723, A237984, A240219, A326622, A339846, A339890, A359910, A362608, A363725, A363727.

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}]

median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);
uniqmode(lista) = { my(freqs=Map(),v); for(i=1,#lista,if(!mapisdefined(freqs,lista[i],&v), v = 0); mapput(freqs,lista[i],1+v)); my(keys=Vec(freqs), fr, mc=0, mf=0, isuniq=0); for(i=1,#keys, fr = mapget(freqs,keys[i]); if(fr>=mf, isuniq = (fr>mf); mf = fr; mc = keys[i])); if(!isuniq, -1, mc); }; \\ Returns -1 if not unique mode.
all_different(facs) = { my(mean=(vecsum(facs)/#facs), med=median(facs), mode=uniqmode(facs)); ((mean!=med) &&  (mean!=mode) && (med!=mode)); };
A363742(n, m=n, facs=List([])) = if(1==n, (#facs>0 && all_different(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A363742(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 29 2025

More terms from Antti Karttunen, Jan 29 2025

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&]

A364058 Heinz numbers of integer partitions with median > 1. Numbers whose multiset of prime factors has median > 2.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, Jul 14 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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:
     3: {2}        23: {9}          42: {1,2,4}
     5: {3}        25: {3,3}        43: {14}
     6: {1,2}      26: {1,6}        45: {2,2,3}
     7: {4}        27: {2,2,2}      46: {1,9}
     9: {2,2}      29: {10}         47: {15}
    10: {1,3}      30: {1,2,3}      49: {4,4}
    11: {5}        31: {11}         50: {1,3,3}
    13: {6}        33: {2,5}        51: {2,7}
    14: {1,4}      34: {1,7}        53: {16}
    15: {2,3}      35: {3,4}        54: {1,2,2,2}
    17: {7}        36: {1,1,2,2}    55: {3,5}
    18: {1,2,2}    37: {12}         57: {2,8}
    19: {8}        38: {1,8}        58: {1,10}
    21: {2,4}      39: {2,6}        59: {17}
    22: {1,5}      41: {13}         60: {1,1,2,3}

For mean instead of median we have A057716, counted by A000065.
These partitions are counted by A238495.
The complement is A364056, counted by A027336, low version A363488.
A000975 counts subsets with integer median, A051293 for mean.
A124943 counts partitions by low median, high version A124944.
A360005 gives twice the median of prime indices, A360459 for prime factors.
A359893 and A359901 count partitions by median.
Cf. A002865, A316413, A325347, A327473, A327476, A363727.

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

A360005(a(n)) > 1.

A360459(a(n)) > 2.

cp's OEIS Frontend

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

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A363721 Number of odd-length integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

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

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A363742 Number of integer factorizations of n with different mean, median, and mode.

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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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A364058 Heinz numbers of integer partitions with median > 1. Numbers whose multiset of prime factors has median > 2.

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