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

2, 12, 18, 36, 120, 270, 360, 540, 600, 750, 1080, 1350, 1500, 1680, 1800, 2250, 2700, 3000, 4500, 5040, 5400, 5670, 6750, 8400, 9000, 11340, 11760, 13500, 15120, 22680, 25200, 26250, 27000, 28350, 35280, 36960, 39690, 42000, 45360, 52500, 56700, 58800, 72030

Offset: 1

Author

Gus Wiseman, Mar 24 2025

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, sum A056239.

Examples

			The terms together with their prime indices begin:
      2: {1}
     12: {1,1,2}
     18: {1,2,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    270: {1,2,2,2,3}
    360: {1,1,1,2,2,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    750: {1,2,3,3,3}
   1080: {1,1,1,2,2,2,3}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   1680: {1,1,1,1,2,3,4}
   1800: {1,1,1,2,2,3,3}
		

Crossrefs

Counting partitions by the LHS gives A008284, rank statistic A061395.

Without the RHS we have A055932, counted by A000009.

Counting partitions by the RHS gives A091602, rank statistic A051903.

Counting partitions by the middle statistic gives A116608/A365676, rank stat A001221.

Without the LHS we have A212166, counted by A239964.

Without the middle statistic we have A381542, counted by A240312.

Partitions of this type are counted by A382302.

A000040 lists the primes, differences A001223.

A001222 counts prime factors, distinct A001221.

A047993 counts balanced partitions, ranks A106529.

A051903 gives greatest prime exponent, least A051904.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represents partition conjugation in terms of Heinz numbers.

Cf. A000720, A048767, A062457, A066328, A130091, A317090, A363719, A363740, A380955, A381632.

Programs

Mathematica

Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==PrimeNu[#]==Max@@FactorInteger[#][[All,2]]&]

Formula

A061395(a(n)) = A001221(a(n)) = A051903(a(n)).