cp's OEIS Frontend

A361856 Positive integers whose prime indices satisfy (maximum) = 2*(median).

%I A361856 #6 Apr 02 2023 09:40:11
%S A361856 12,24,42,48,60,63,72,96,126,130,140,144,189,192,195,252,266,288,308,
%T A361856 325,330,360,378,384,399,420,432,495,546,567,572,576,588,600,630,638,
%U A361856 650,665,756,768,819,864,882,884,931,945,957,962,975,1122,1134,1152,1190
%N A361856 Positive integers whose prime indices satisfy (maximum) = 2*(median).
%C A361856 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 A361856 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%C A361856 These are Heinz numbers of partitions satisfying (maximum) = 2*(median).
%F A361856 A061395(a(n)) = 2*A360005(a(n)).
%e A361856 The terms together with their prime indices begin:
%e A361856     12: {1,1,2}
%e A361856     24: {1,1,1,2}
%e A361856     42: {1,2,4}
%e A361856     48: {1,1,1,1,2}
%e A361856     60: {1,1,2,3}
%e A361856     63: {2,2,4}
%e A361856     72: {1,1,1,2,2}
%e A361856     96: {1,1,1,1,1,2}
%e A361856    126: {1,2,2,4}
%e A361856    130: {1,3,6}
%e A361856    140: {1,1,3,4}
%e A361856    144: {1,1,1,1,2,2}
%e A361856 The prime indices of 126 are {1,2,2,4}, with maximum 4 and median 2, so 126 is in the sequence.
%e A361856 The prime indices of 308 are {1,1,4,5}, with maximum 5 and median 5/2, so 308 is in the sequence.
%t A361856 prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A361856 Select[Range[100],Max@@prix[#]==2*Median[prix[#]]&]
%Y A361856 The LHS (greatest prime index) is A061395.
%Y A361856 The RHS (twice median) is A360005, distinct A360457.
%Y A361856 These partitions are counted by A361849.
%Y A361856 For mean instead of median we have A361855, counted by A361853.
%Y A361856 For minimum instead of median we have A361908, counted by A118096.
%Y A361856 For length instead of median we have A361909, counted by A237753.
%Y A361856 A000975 counts subsets with integer median.
%Y A361856 A001222 (bigomega) counts prime factors, distinct A001221 (omega).
%Y A361856 A112798 lists prime indices, sum A056239.
%Y A361856 A325347 counts partitions with integer median, complement A307683.
%Y A361856 A359893 and A359901 count partitions by median.
%Y A361856 Cf. A027193, A067801, A111907, A361205, A361848, A361858.
%K A361856 nonn
%O A361856 1,1
%A A361856 _Gus Wiseman_, Apr 02 2023