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A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

%I A381543 #7 Mar 24 2025 22:35:15
%S A381543 2,12,18,36,120,270,360,540,600,750,1080,1350,1500,1680,1800,2250,
%T A381543 2700,3000,4500,5040,5400,5670,6750,8400,9000,11340,11760,13500,15120,
%U A381543 22680,25200,26250,27000,28350,35280,36960,39690,42000,45360,52500,56700,58800,72030
%N A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.
%C A381543 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, sum A056239.
%F A381543 A061395(a(n)) = A001221(a(n)) = A051903(a(n)).
%e A381543 The terms together with their prime indices begin:
%e A381543       2: {1}
%e A381543      12: {1,1,2}
%e A381543      18: {1,2,2}
%e A381543      36: {1,1,2,2}
%e A381543     120: {1,1,1,2,3}
%e A381543     270: {1,2,2,2,3}
%e A381543     360: {1,1,1,2,2,3}
%e A381543     540: {1,1,2,2,2,3}
%e A381543     600: {1,1,1,2,3,3}
%e A381543     750: {1,2,3,3,3}
%e A381543    1080: {1,1,1,2,2,2,3}
%e A381543    1350: {1,2,2,2,3,3}
%e A381543    1500: {1,1,2,3,3,3}
%e A381543    1680: {1,1,1,1,2,3,4}
%e A381543    1800: {1,1,1,2,2,3,3}
%t A381543 Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==PrimeNu[#]==Max@@FactorInteger[#][[All,2]]&]
%Y A381543 Counting partitions by the LHS gives A008284, rank statistic A061395.
%Y A381543 Without the RHS we have A055932, counted by A000009.
%Y A381543 Counting partitions by the RHS gives A091602, rank statistic A051903.
%Y A381543 Counting partitions by the middle statistic gives A116608/A365676, rank stat A001221.
%Y A381543 Without the LHS we have A212166, counted by A239964.
%Y A381543 Without the middle statistic we have A381542, counted by A240312.
%Y A381543 Partitions of this type are counted by A382302.
%Y A381543 A000040 lists the primes, differences A001223.
%Y A381543 A001222 counts prime factors, distinct A001221.
%Y A381543 A047993 counts balanced partitions, ranks A106529.
%Y A381543 A051903 gives greatest prime exponent, least A051904.
%Y A381543 A055396 gives least prime index, greatest A061395.
%Y A381543 A056239 adds up prime indices, row sums of A112798.
%Y A381543 A122111 represents partition conjugation in terms of Heinz numbers.
%Y A381543 Cf. A000720, A048767, A062457, A066328, A130091, A317090, A363719, A363740, A380955, A381632.
%K A381543 nonn
%O A381543 1,1
%A A381543 _Gus Wiseman_, Mar 24 2025