A344292 Numbers m whose sum of prime indices A056239(m) is even and is at most twice the number of prime factors counted with multiplicity A001222(m).

1, 3, 4, 9, 10, 12, 16, 27, 28, 30, 36, 40, 48, 64, 81, 84, 88, 90, 100, 108, 112, 120, 144, 160, 192, 208, 243, 252, 256, 264, 270, 280, 300, 324, 336, 352, 360, 400, 432, 448, 480, 544, 576, 624, 640, 729, 756, 768, 784, 792, 810, 832, 840, 880, 900, 972

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.

Also Heinz numbers of integer partitions of even numbers m with at least m/2 parts, counted by A000070 riffled with 0's, or A025065 with odd positions zeroed out.

Links

Table of n, a(n) for n=1..56.

Formula

Members m of A300061 such that A056239(m) <= 2*A001222(m).

Example

The sequence of terms together with their prime indices begins:
       1: {}                 84: {1,1,2,4}
       3: {2}                88: {1,1,1,5}
       4: {1,1}              90: {1,2,2,3}
       9: {2,2}             100: {1,1,3,3}
      10: {1,3}             108: {1,1,2,2,2}
      12: {1,1,2}           112: {1,1,1,1,4}
      16: {1,1,1,1}         120: {1,1,1,2,3}
      27: {2,2,2}           144: {1,1,1,1,2,2}
      28: {1,1,4}           160: {1,1,1,1,1,3}
      30: {1,2,3}           192: {1,1,1,1,1,1,2}
      36: {1,1,2,2}         208: {1,1,1,1,6}
      40: {1,1,1,3}         243: {2,2,2,2,2}
      48: {1,1,1,1,2}       252: {1,1,2,2,4}
      64: {1,1,1,1,1,1}     256: {1,1,1,1,1,1,1,1}
      81: {2,2,2,2}         264: {1,1,1,2,5}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]>=Total[primeMS[#]]/2&]

Crossrefs

These are the Heinz numbers of partitions counted by A000070 and A025065.

A subset of A300061 (sum of prime indices is even).

The conjugate opposite version is A320924, counted by A209816.

The conjugate opposite version allowing odds is A322109, counted by A110618.

The case of equality is A340387, counted by A000041.

The opposite version allowing odd weights is A344291, counted by A110618.

Allowing odd weights gives A344296, counted by A025065.

The opposite version is A344413, counted by A209816.

The conjugate version allowing odd weights is A344414, counted by A025065.

The case of equality in the conjugate case is A344415, counted by A035363.

The conjugate version is A344416, counted by A000070.

A001222 counts prime factors with multiplicity.

A027187 counts partitions of even length, ranked by A028260.

A056239 adds up prime indices, row sums of A112798.

A058696 counts partitions of even numbers, ranked by A300061.

A301987 lists numbers whose sum of prime indices equals their product.

A330950 counts partitions of n with Heinz number divisible by n.

A334201 adds up all prime indices except the greatest.

Cf. A001414, A067538, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.

Sequence in context: A336226 A339658 A344297 * A356823 A345359 A377430

Adjacent sequences: A344289 A344290 A344291 * A344293 A344294 A344295

Keyword

nonn

Author

Gus Wiseman, May 22 2021

Status

approved