A361908 Positive integers > 1 whose prime indices satisfy (maximum) = 2*(minimum).

6, 12, 18, 21, 24, 36, 48, 54, 63, 65, 72, 96, 105, 108, 133, 144, 147, 162, 189, 192, 216, 288, 315, 319, 324, 325, 384, 432, 441, 455, 481, 486, 525, 567, 576, 648, 715, 731, 735, 768, 845, 864, 931, 945, 972, 1007, 1029, 1152, 1296, 1323, 1403, 1458, 1463

Offset: 1

Author

Gus Wiseman, Apr 05 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.

Examples

			The terms together with their prime indices begin:
     6: {1,2}
    12: {1,1,2}
    18: {1,2,2}
    21: {2,4}
    24: {1,1,1,2}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    63: {2,2,4}
    65: {3,6}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

The RHS is 2*A055396 (twice minimum).

The LHS is A061395 (greatest prime index).

Partitions of this type are counted by A118096.

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

For median instead of minimum we have A361856, counted by A361849.

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

A001221 (omega) counts distinct prime factors.

A001222 (bigomega) counts prime factors with multiplicity.

A112798 lists prime indices, sum A056239.

Cf. A053263, A067801, A237820, A237821, A361858.

Programs

Maple

filter:= proc(n) local F,b;
  if n::even then b:= padic:-ordp(n,3);
     if b = 0 then return false else return n = 2^padic:-ordp(n,2) * 3^b fi
  fi;
  F:= ifactors(n)[2][..,1];
  nops(F) >= 2 and numtheory:-pi(max(F)) = 2*numtheory:-pi(min(F))
end proc:
select(filter, [$1..2000]); # Robert Israel, Mar 11 2025

Mathematica

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]==2*PrimePi[FactorInteger[#][[1,1]]]&]