A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).

1, 3, 5, 9, 10, 15, 25, 27, 30, 45, 50, 75, 81, 90, 100, 125, 135, 150, 225, 243, 250, 270, 300, 375, 405, 450, 500, 625, 675, 729, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050

Offset: 1

Author

Gus Wiseman, May 16 2021

Keywords

• nonn

Comments

A number is 5-smooth if its prime divisors are all <= 5.

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 sequence of terms together with their prime indices begins:
       1: {}            125: {3,3,3}
       3: {2}           135: {2,2,2,3}
       5: {3}           150: {1,2,3,3}
       9: {2,2}         225: {2,2,3,3}
      10: {1,3}         243: {2,2,2,2,2}
      15: {2,3}         250: {1,3,3,3}
      25: {3,3}         270: {1,2,2,2,3}
      27: {2,2,2}       300: {1,1,2,3,3}
      30: {1,2,3}       375: {2,3,3,3}
      45: {2,2,3}       405: {2,2,2,2,3}
      50: {1,3,3}       450: {1,2,2,3,3}
      75: {2,3,3}       500: {1,1,3,3,3}
      81: {2,2,2,2}     625: {3,3,3,3}
      90: {1,2,2,3}     675: {2,2,2,3,3}
     100: {1,1,3,3}     729: {2,2,2,2,2,2}
		

Crossrefs

Allowing any number of parts and sum gives A051037, counted by A001399.

These are Heinz numbers of the partitions counted by A266755.

Allowing parts > 5 gives A344291, counted by A110618.

The non-3-smooth case is A344294, counted by A325691.

Requiring the sum of prime indices to be even gives A344295.

A000070 counts non-multigraphical partitions, ranked by A344292.

A025065 counts partitions of n with >= n/2 parts, ranked by A344296.

A035363 counts partitions of n with n/2 parts, ranked by A340387.

A056239 adds up prime indices, row sums of A112798.

A300061 ranks partitions of even numbers, with 5-smooth case A344297.

Cf. A000041, A000244, A026811, A080193, A244990, A261144, A279622.

Programs

Mathematica

Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]<=5&]

Formula

Intersection of A051037 and A344291.