A344296 Numbers with at least as many prime factors (counted with multiplicity) as half their sum of prime indices.

1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 256, 264, 270, 280, 288, 300, 320, 324, 336, 352

Offset

1,2

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.

These are the Heinz numbers of certain partitions counted by A025065, but different from palindromic partitions, which have Heinz numbers A265640.

Links

Table of n, a(n) for n=1..60.

Formula

A056239(a(n)) <= 2*A001222(a(n)).

a(n) = A322136(n)/4.

Example

The sequence of terms together with their prime indices begins:
      1: {}            30: {1,2,3}
      2: {1}           32: {1,1,1,1,1}
      3: {2}           36: {1,1,2,2}
      4: {1,1}         40: {1,1,1,3}
      6: {1,2}         48: {1,1,1,1,2}
      8: {1,1,1}       54: {1,2,2,2}
      9: {2,2}         56: {1,1,1,4}
     10: {1,3}         60: {1,1,2,3}
     12: {1,1,2}       64: {1,1,1,1,1,1}
     16: {1,1,1,1}     72: {1,1,1,2,2}
     18: {1,2,2}       80: {1,1,1,1,3}
     20: {1,1,3}       81: {2,2,2,2}
     24: {1,1,1,2}     84: {1,1,2,4}
     27: {2,2,2}       88: {1,1,1,5}
     28: {1,1,4}       90: {1,2,2,3}

Mathematica

Select[Range[100], PrimeOmega[#]>=Total[Cases[FactorInteger[#], {p_, k_}:>k*PrimePi[p]]]/2&]

Crossrefs

The case with difference at least 1 is A322136.

The case of equality is A340387, counted by A000041 or A035363.

The opposite version is A344291, counted by A110618.

The conjugate version is A344414, with even-weight case A344416.

A025065 counts palindromic partitions, ranked by A265640.

A056239 adds up prime indices, row sums of A112798.

A300061 lists numbers whose sum of prime indices is even.

Cf. A001399, A002865, A025147, A027336, A036036, A067712, A244990, A261144, A325691, A344293, A344295.

Sequence in context: A269045 A067023 A227762 * A333506 A303492 A188591

Adjacent sequences: A344293 A344294 A344295 * A344297 A344298 A344299

Keyword

nonn

Author

Gus Wiseman, May 16 2021

Status

approved