cp's OEIS Frontend

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

%I A360249 #8 May 22 2023 05:58:10
%S A360249 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 A360249 32,33,34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,
%U A360249 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 A360249 Numbers for which the prime indices have the same median as the distinct prime indices.
%C A360249 First differs from A072774 in having 90.
%C A360249 First differs from A242414 in having 180.
%C A360249 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 A360249 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 A360249 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 A360249 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 A360249 isA360249 := proc(n)
%p A360249     local ifs,pidx,pe,medAll,medDist ;
%p A360249     if n = 1 then
%p A360249         return true ;
%p A360249     end if ;
%p A360249     ifs := ifactors(n)[2] ;
%p A360249     pidx := [] ;
%p A360249     for pe in ifs do
%p A360249         numtheory[pi](op(1,pe)) ;
%p A360249         pidx := [op(pidx),seq(%,i=1..op(2,pe))] ;
%p A360249     end do:
%p A360249     medAll := stats[describe,median](sort(pidx)) ;
%p A360249     pidx := convert(convert(pidx,set),list) ;
%p A360249     medDist := stats[describe,median](sort(pidx)) ;
%p A360249     if medAll = medDist then
%p A360249         true;
%p A360249     else
%p A360249         false;
%p A360249     end if;
%p A360249 end proc:
%p A360249 for n from 1 to 130 do
%p A360249     if isA360249(n) then
%p A360249         printf("%d,",n) ;
%p A360249     end if;
%p A360249 end do: # _R. J. Mathar_, May 22 2023
%t A360249 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A360249 Select[Range[100],Median[prix[#]]==Median[Union[prix[#]]]&]
%Y A360249 These partitions are counted by A360245.
%Y A360249 The complement for mean instead of median is A360246, counted by A360242.
%Y A360249 For mean instead of median we have A360247, counted by A360243.
%Y A360249 The complement is A360248, counted by A360244.
%Y A360249 For multiplicities instead of parts: A360453, counted by A360455.
%Y A360249 For multiplicities instead of distinct parts: A360454, counted by A360456.
%Y A360249 A112798 lists prime indices, length A001222, sum A056239.
%Y A360249 A240219 counts partitions with mean equal to median, ranks A359889.
%Y A360249 A326567/A326568 gives mean of prime indices.
%Y A360249 A326619/A326620 gives mean of distinct prime indices.
%Y A360249 A325347 = partitions with integer median, strict A359907, ranks A359908.
%Y A360249 A359893 and A359901 count partitions by median.
%Y A360249 A359894 = partitions with mean different from median, ranks A359890.
%Y A360249 A360005 gives median of prime indices (times two).
%Y A360249 Cf. A000975, A078174, A316413, A324570, A359903, A360252, A360253.
%K A360249 nonn
%O A360249 1,2
%A A360249 _Gus Wiseman_, Feb 07 2023