A360247 Numbers for which the prime indices have the same mean 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, 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 lacking 126.

Includes all squarefree numbers and perfect powers.

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.

Examples

			The prime indices of 900 are {3,3,2,2,1,1} with mean 2, and the distinct prime indices are {1,2,3} also with mean 2, so 900 is in the sequence.
		

Crossrefs

Signature instead of parts: A324570, counted by A114638.

Signature instead of distinct parts: A359903, counted by A360068.

These partitions are counted by A360243.

The complement is A360246, counted by A360242.

For median instead of mean the complement is A360248, counted by A360244.

For median instead of mean we have A360249, counted by A360245.

For greater instead of equal mean we have A360252, counted by A360250.

For lesser instead of equal mean we have A360253, counted by A360251.

A008284 counts partitions by number of parts, distinct A116608.

A058398 counts partitions by mean, also A327482.

A088529/A088530 gives mean of prime signature (A124010).

A112798 lists prime indices, length A001222, sum A056239.

A316413 = numbers whose prime indices have integer mean, distinct A326621.

A326567/A326568 gives mean of prime indices.

A326619/A326620 gives mean of distinct prime indices.

Cf. A000975, A051293, A067340, A067538, A360005, A360241.

Programs

Maple

isA360247 := proc(n)
    local ifs,pidx,pe,meanAll,meanDist ;
    if n = 1 then
        return true ;
    end if ;
    ifs := ifactors(n)[2] ;
    # list of prime indices with multiplicity
    pidx := [] ;
    for pe in ifs do
        [numtheory[pi](op(1,pe)),op(2,pe)] ;
        pidx := [op(pidx),%] ;
    end do:
    meanAll := add(op(1,pe)*op(2,pe),pe=pidx) / add(op(2,pe),pe=pidx) ;
    meanDist := add(op(1,pe),pe=pidx) / nops(pidx) ;
    if meanAll = meanDist then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 130 do
    if isA360247(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],Mean[prix[#]]==Mean[Union[prix[#]]]&]