A360249 Numbers for which the prime indices have the same median as the distinct prime indices.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 121, 122, 123, 125, 126, 127, 128, 129, 130

Offset: 1

Author

Gus Wiseman, Feb 07 2023

Keywords

• nonn

Comments

First differs from A072774 in having 90.

First differs from A242414 in having 180.

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).

Examples

			The prime indices of 126 are {1,2,2,4} with median 2 and distinct prime indices {1,2,4} with median 2, so 126 is in the sequence.
The prime indices of 180 are {1,1,2,2,3} with median 2 and distinct prime indices {1,2,3} with median 2, so 180 is in the sequence.
		

Crossrefs

These partitions are counted by A360245.

The complement for mean instead of median is A360246, counted by A360242.

For mean instead of median we have A360247, counted by A360243.

The complement is A360248, counted by A360244.

For multiplicities instead of parts: A360453, counted by A360455.

For multiplicities instead of distinct parts: A360454, counted by A360456.

A112798 lists prime indices, length A001222, sum A056239.

A240219 counts partitions with mean equal to median, ranks A359889.

A326567/A326568 gives mean of prime indices.

A326619/A326620 gives mean of distinct prime indices.

A325347 = partitions with integer median, strict A359907, ranks A359908.

A359893 and A359901 count partitions by median.

A359894 = partitions with mean different from median, ranks A359890.

A360005 gives median of prime indices (times two).

Cf. A000975, A078174, A316413, A324570, A359903, A360252, A360253.

Programs

Maple

isA360249 := proc(n)
    local ifs,pidx,pe,medAll,medDist ;
    if n = 1 then
        return true ;
    end if ;
    ifs := ifactors(n)[2] ;
    pidx := [] ;
    for pe in ifs do
        numtheory[pi](op(1,pe)) ;
        pidx := [op(pidx),seq(%,i=1..op(2,pe))] ;
    end do:
    medAll := stats[describe,median](sort(pidx)) ;
    pidx := convert(convert(pidx,set),list) ;
    medDist := stats[describe,median](sort(pidx)) ;
    if medAll = medDist then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 130 do
    if isA360249(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 22 2023

Mathematica

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