A363740 -id:A363740 - cp's OEIS frontend

A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

90, 270, 525, 550, 756, 810, 1666, 1911, 1950, 2268, 2430, 2625, 2695, 2700, 2750, 5566, 6762, 6804, 6897, 7128, 7290, 8100, 8500, 9310, 9750, 10285, 10478, 11011, 11550, 11662, 12250, 12375, 12495, 13125, 13377, 13750, 14014, 14703, 18865, 19435, 20412, 21384

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 prime indices of 6897 are {2,5,5,8}, with mean 5, median 5, and modes {5}, so 6897 is in the sequence.
The terms together with their prime indices begin:
     90: {1,2,2,3}
    270: {1,2,2,2,3}
    525: {2,3,3,4}
    550: {1,3,3,5}
    756: {1,1,2,2,2,4}
    810: {1,2,2,2,2,3}
   1666: {1,4,4,7}
   1911: {2,4,4,6}
   1950: {1,2,3,3,6}
   2268: {1,1,2,2,2,2,4}
   2430: {1,2,2,2,2,2,3}

For just primes instead of prime powers we have A363722.
Including prime-powers gives A363727, counted by A363719.
These partitions are counted by A363728.
For unequal instead of equal we have A363730, counted by A363720.
A000961 lists the prime powers, complement A024619.
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. A215366, A327473, A327476, A359893, A359908, A360009, A360248, 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[1000],!PrimePowerQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

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

Original entry on oeis.org

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}]

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}]

A363726 Number of odd-length integer partitions of n with a unique mode.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 22, 26, 39, 50, 67, 86, 118, 148, 196, 245, 315, 394, 507, 629, 792, 979, 1231, 1503, 1873, 2286, 2814, 3424, 4194, 5073, 6183, 7449, 9014, 10827, 13055, 15603, 18713, 22308, 26631, 31646, 37641, 44559, 52835, 62374, 73671

Offset: 0

Gus Wiseman, Jun 27 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 a(1) = 1 through a(8) = 8 partitions:
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)
            (111)  (211)  (221)    (222)    (322)      (332)
                          (311)    (411)    (331)      (422)
                          (11111)  (21111)  (511)      (611)
                                            (22111)    (22211)
                                            (31111)    (32111)
                                            (1111111)  (41111)
                                                       (2111111)

The constant case is A001227.
Allowing multiple modes gives A027193, ranks A026424.
Allowing any length gives A362608, ranks A356862.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median.
Cf. A325347, A363719, A363720, A363731, A363724, A363725, A363740.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], Length[modes[#]]==1&&OddQ[Length[#]]&]],{n,30}]

A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

Original entry on oeis.org

2, 12, 18, 36, 120, 270, 360, 540, 600, 750, 1080, 1350, 1500, 1680, 1800, 2250, 2700, 3000, 4500, 5040, 5400, 5670, 6750, 8400, 9000, 11340, 11760, 13500, 15120, 22680, 25200, 26250, 27000, 28350, 35280, 36960, 39690, 42000, 45360, 52500, 56700, 58800, 72030

Offset: 1

Gus Wiseman, Mar 24 2025

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, sum A056239.

			The terms together with their prime indices begin:
      2: {1}
     12: {1,1,2}
     18: {1,2,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    270: {1,2,2,2,3}
    360: {1,1,1,2,2,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    750: {1,2,3,3,3}
   1080: {1,1,1,2,2,2,3}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   1680: {1,1,1,1,2,3,4}
   1800: {1,1,1,2,2,3,3}

Counting partitions by the LHS gives A008284, rank statistic A061395.
Without the RHS we have A055932, counted by A000009.
Counting partitions by the RHS gives A091602, rank statistic A051903.
Counting partitions by the middle statistic gives A116608/A365676, rank stat A001221.
Without the LHS we have A212166, counted by A239964.
Without the middle statistic we have A381542, counted by A240312.
Partitions of this type are counted by A382302.
A000040 lists the primes, differences A001223.
A001222 counts prime factors, distinct A001221.
A047993 counts balanced partitions, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents partition conjugation in terms of Heinz numbers.
Cf. A000720, A048767, A062457, A066328, A130091, A317090, A363719, A363740, A380955, A381632.

Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==PrimeNu[#]==Max@@FactorInteger[#][[All,2]]&]

A061395(a(n)) = A001221(a(n)) = A051903(a(n)).

A362562 Number of non-constant integer partitions of n having a unique mode equal to the mean.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 3, 3, 7, 0, 12, 0, 18, 12, 9, 0, 52, 12, 14, 33, 54, 0, 121, 0, 98, 76, 31, 100, 343, 0, 45, 164, 493, 0, 548, 0, 483, 757, 88, 0, 1789, 289, 979, 645, 1290, 0, 2225, 1677, 3371, 1200, 221, 0, 10649

Offset: 0

Gus Wiseman, Jun 27 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 a(8) = 1 through a(16) = 7 partitions:
  (3221)  .  (32221)  .  (4332)    .  (3222221)  (43332)  (5443)
                         (5331)       (3322211)  (53331)  (6442)
                         (322221)     (4222211)  (63321)  (7441)
                         (422211)                         (32222221)
                                                          (33222211)
                                                          (42222211)
                                                          (52222111)

Partitions containing their mean are counted by A237984, ranks A327473.
Partitions missing their mean are counted by A327472, ranks A327476.
Allowing constant partitions gives A363723.
Including median also gives A363728, ranks A363729.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median.
A362608 counts partitions with a unique mode.
Cf. A240219, A325347, A363719, A363720, A363724, A363725, A363731, A363740.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&{Mean[#]}==modes[#]&]],{n,0,30}]

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)).

A364192 High (i.e., greatest) co-mode in the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 16 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 define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.
Extending the terminology of A124943, the "high co-mode" in a multiset is the greatest co-mode.

			The prime indices of 2100 are {1,1,2,3,3,4}, with co-modes {2,4}, so a(2100) = 4.

For prime factors instead of indices we have A359612, low A067695.
For mode instead of co-mode we have A363487 (triangle A363953), low A363486 (triangle A363952).
The version for median instead of co-mode is A363942, low A363941.
Positions of 1's are A364061, counted by A364062.
The low version is A364191, 1's at A364158 (counted by A364159).
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A241131, A327473, A327476, A360005, A360015, A362612, A363740.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==1,0,Max[comodes[prix[n]]]],{n,30}]

a(n) = A000720(A359612(n)).

A359612(n) = A000040(a(n)).

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&]

cp's OEIS Frontend

A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

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

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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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A363726 Number of odd-length integer partitions of n with a unique mode.

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A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

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A362562 Number of non-constant integer partitions of n having a unique mode equal to the mean.

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

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A364192 High (i.e., greatest) co-mode in the multiset of prime indices of n.

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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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