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

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

Author

Gus Wiseman, Jun 24 2023

Keywords

• nonn

Comments

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).

Examples

			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}
		

Crossrefs

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.

Programs

Mathematica

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[#]]&]

Formula

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)).