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

12, 24, 42, 48, 60, 63, 72, 96, 126, 130, 140, 144, 189, 192, 195, 252, 266, 288, 308, 325, 330, 360, 378, 384, 399, 420, 432, 495, 546, 567, 572, 576, 588, 600, 630, 638, 650, 665, 756, 768, 819, 864, 882, 884, 931, 945, 957, 962, 975, 1122, 1134, 1152, 1190

Offset: 1

Author

Gus Wiseman, Apr 02 2023

Keywords

• nonn

Comments

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.

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

These are Heinz numbers of partitions satisfying (maximum) = 2*(median).

Examples

			The terms together with their prime indices begin:
    12: {1,1,2}
    24: {1,1,1,2}
    42: {1,2,4}
    48: {1,1,1,1,2}
    60: {1,1,2,3}
    63: {2,2,4}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
   126: {1,2,2,4}
   130: {1,3,6}
   140: {1,1,3,4}
   144: {1,1,1,1,2,2}
The prime indices of 126 are {1,2,2,4}, with maximum 4 and median 2, so 126 is in the sequence.
The prime indices of 308 are {1,1,4,5}, with maximum 5 and median 5/2, so 308 is in the sequence.
		

Crossrefs

The LHS (greatest prime index) is A061395.

The RHS (twice median) is A360005, distinct A360457.

These partitions are counted by A361849.

For mean instead of median we have A361855, counted by A361853.

For minimum instead of median we have A361908, counted by A118096.

For length instead of median we have A361909, counted by A237753.

A000975 counts subsets with integer median.

A001222 (bigomega) counts prime factors, distinct A001221 (omega).

A112798 lists prime indices, sum A056239.

A325347 counts partitions with integer median, complement A307683.

A359893 and A359901 count partitions by median.

Cf. A027193, A067801, A111907, A361205, A361848, A361858.

Programs

Mathematica

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@prix[#]==2*Median[prix[#]]&]

Formula

A061395(a(n)) = 2*A360005(a(n)).