cp's OEIS Frontend

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

%I A361908 #10 Mar 12 2025 08:15:04
%S A361908 6,12,18,21,24,36,48,54,63,65,72,96,105,108,133,144,147,162,189,192,
%T A361908 216,288,315,319,324,325,384,432,441,455,481,486,525,567,576,648,715,
%U A361908 731,735,768,845,864,931,945,972,1007,1029,1152,1296,1323,1403,1458,1463
%N A361908 Positive integers > 1 whose prime indices satisfy (maximum) = 2*(minimum).
%C A361908 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.
%H A361908 Robert Israel, <a href="/A361908/b361908.txt">Table of n, a(n) for n = 1..10000</a>
%e A361908 The terms together with their prime indices begin:
%e A361908      6: {1,2}
%e A361908     12: {1,1,2}
%e A361908     18: {1,2,2}
%e A361908     21: {2,4}
%e A361908     24: {1,1,1,2}
%e A361908     36: {1,1,2,2}
%e A361908     48: {1,1,1,1,2}
%e A361908     54: {1,2,2,2}
%e A361908     63: {2,2,4}
%e A361908     65: {3,6}
%e A361908     72: {1,1,1,2,2}
%e A361908     96: {1,1,1,1,1,2}
%p A361908 filter:= proc(n) local F,b;
%p A361908   if n::even then b:= padic:-ordp(n,3);
%p A361908      if b = 0 then return false else return n = 2^padic:-ordp(n,2) * 3^b fi
%p A361908   fi;
%p A361908   F:= ifactors(n)[2][..,1];
%p A361908   nops(F) >= 2 and numtheory:-pi(max(F)) = 2*numtheory:-pi(min(F))
%p A361908 end proc:
%p A361908 select(filter, [$1..2000]); # _Robert Israel_, Mar 11 2025
%t A361908 Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]==2*PrimePi[FactorInteger[#][[1,1]]]&]
%Y A361908 The RHS is 2*A055396 (twice minimum).
%Y A361908 The LHS is A061395 (greatest prime index).
%Y A361908 Partitions of this type are counted by A118096.
%Y A361908 For mean instead of minimum we have A361855, counted by A361853.
%Y A361908 For median instead of minimum we have A361856, counted by A361849.
%Y A361908 For length instead of minimum we have A361909, counted by A237753.
%Y A361908 A001221 (omega) counts distinct prime factors.
%Y A361908 A001222 (bigomega) counts prime factors with multiplicity.
%Y A361908 A112798 lists prime indices, sum A056239.
%Y A361908 Cf. A053263, A067801, A237820, A237821, A361858.
%K A361908 nonn
%O A361908 1,1
%A A361908 _Gus Wiseman_, Apr 05 2023