%I #7 May 22 2023 05:43:18

%S 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,

%T 32,33,34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,

%U 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

%N Numbers for which the prime indices have the same mean as the distinct prime indices.

%C First differs from A072774 in having 90.

%C First differs from A242414 in lacking 126.

%C Includes all squarefree numbers and perfect powers.

%C 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.

%e 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.

%p isA360247 := proc(n)

%p local ifs,pidx,pe,meanAll,meanDist ;

%p if n = 1 then

%p return true ;

%p end if ;

%p ifs := ifactors(n)[2] ;

%p # list of prime indices with multiplicity

%p pidx := [] ;

%p for pe in ifs do

%p [numtheory[pi](op(1,pe)),op(2,pe)] ;

%p pidx := [op(pidx),%] ;

%p end do:

%p meanAll := add(op(1,pe)*op(2,pe),pe=pidx) / add(op(2,pe),pe=pidx) ;

%p meanDist := add(op(1,pe),pe=pidx) / nops(pidx) ;

%p if meanAll = meanDist then

%p true;

%p else

%p false;

%p end if;

%p end proc:

%p for n from 1 to 130 do

%p if isA360247(n) then

%p printf("%d,",n) ;

%p end if;

%p end do: # _R. J. Mathar_, May 22 2023

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Mean[prix[#]]==Mean[Union[prix[#]]]&]

%Y Signature instead of parts: A324570, counted by A114638.

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

%Y These partitions are counted by A360243.

%Y The complement is A360246, counted by A360242.

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

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

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

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

%Y A008284 counts partitions by number of parts, distinct A116608.

%Y A058398 counts partitions by mean, also A327482.

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

%Y A112798 lists prime indices, length A001222, sum A056239.

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

%Y A326567/A326568 gives mean of prime indices.

%Y A326619/A326620 gives mean of distinct prime indices.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 07 2023