A360249 - OEIS

%I #8 May 22 2023 05:58:10

%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 65,66,67,69,70,71,73,74,77,78,79,81,82,83,85,86,87,89,90,91,93,94,95,97,100,101,102,103,105,106,107,109,110,111,113,114,115,118,119,121,122,123,125,126,127,128,129,130

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

%C First differs from A072774 in having 90.

%C First differs from A242414 in having 180.

%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.

%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

%e 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.

%e 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.

%p isA360249 := proc(n)

%p     local ifs,pidx,pe,medAll,medDist ;

%p     if n = 1 then

%p         return true ;

%p     end if ;

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

%p     pidx := [] ;

%p     for pe in ifs do

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

%p         pidx := [op(pidx),seq(%,i=1..op(2,pe))] ;

%p     end do:

%p     medAll := stats[describe,median](sort(pidx)) ;

%p     pidx := convert(convert(pidx,set),list) ;

%p     medDist := stats[describe,median](sort(pidx)) ;

%p     if medAll = medDist then

%p         true;

%p     else

%p         false;

%p     end if;

%p end proc:

%p for n from 1 to 130 do

%p     if isA360249(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],Median[prix[#]]==Median[Union[prix[#]]]&]

%Y These partitions are counted by A360245.

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

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

%Y The complement is A360248, counted by A360244.

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

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

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

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

%Y A326567/A326568 gives mean of prime indices.

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

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

%Y A359893 and A359901 count partitions by median.

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

%Y A360005 gives median of prime indices (times two).

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 07 2023