A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

5, 10, 15, 25, 30, 45, 50, 75, 90, 100, 125, 135, 150, 225, 250, 270, 300, 375, 405, 450, 500, 625, 675, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050, 4500, 5000, 5625, 6075, 6250

Offset: 1

Author

Gus Wiseman, May 16 2021

Keywords

• nonn

Comments

A number is d-smooth iff its prime divisors are all <= d.

A prime index of k is a number m such that prime(m) divides k, and the multiset of prime indices of k is row k of A112798. This row has length A001222(k) and sum A056239(k).

Examples

			The sequence of terms together with their prime indices begins:
       5: {3}           270: {1,2,2,2,3}
      10: {1,3}         300: {1,1,2,3,3}
      15: {2,3}         375: {2,3,3,3}
      25: {3,3}         405: {2,2,2,2,3}
      30: {1,2,3}       450: {1,2,2,3,3}
      45: {2,2,3}       500: {1,1,3,3,3}
      50: {1,3,3}       625: {3,3,3,3}
      75: {2,3,3}       675: {2,2,2,3,3}
      90: {1,2,2,3}     750: {1,2,3,3,3}
     100: {1,1,3,3}     810: {1,2,2,2,2,3}
     125: {3,3,3}       900: {1,1,2,2,3,3}
     135: {2,2,2,3}    1000: {1,1,1,3,3,3}
     150: {1,2,3,3}    1125: {2,2,3,3,3}
     225: {2,2,3,3}    1215: {2,2,2,2,2,3}
     250: {1,3,3,3}    1250: {1,3,3,3,3}
		

Crossrefs

Allowing any number of parts and sum gives A080193, counted by A069905.

The partitions with these Heinz numbers are counted by A325691.

Relaxing the smoothness conditions gives A344291, counted by A110618.

Allowing 3-smoothness gives A344293, counted by A266755.

A025065 counts partitions of n with at least n/2 parts, ranked by A344296.

A035363 counts partitions of n whose length is n/2, ranked by A340387.

A051037 lists 5-smooth numbers (complement: A279622).

A056239 adds up prime indices, row sums of A112798.

A257993 gives the least gap of the partition with Heinz number n.

A300061 lists numbers with even sum of prime indices (5-smooth: A344297).

A342050/A342051 list Heinz numbers of partitions with even/odd least gap.

Cf. A000041, A001399, A002865, A027336, A244990, A261144, A344295.

Programs

Mathematica

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

Formula

Intersection of A080193 and A344291.