A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).

1, 3, 7, 9, 10, 13, 19, 21, 22, 25, 27, 28, 29, 30, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138

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 most m/2 parts, counted by A209816 riffled with zeros, or A110618 with odd positions zeroed out.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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

Example

The sequence of terms together with their prime indices begins:
      1: {}          37: {12}        75: {2,3,3}
      3: {2}         39: {2,6}       76: {1,1,8}
      7: {4}         43: {14}        79: {22}
      9: {2,2}       46: {1,9}       81: {2,2,2,2}
     10: {1,3}       49: {4,4}       82: {1,13}
     13: {6}         52: {1,1,6}     84: {1,1,2,4}
     19: {8}         53: {16}        85: {3,7}
     21: {2,4}       55: {3,5}       87: {2,10}
     22: {1,5}       57: {2,8}       88: {1,1,1,5}
     25: {3,3}       61: {18}        89: {24}
     27: {2,2,2}     62: {1,11}      90: {1,2,2,3}
     28: {1,1,4}     63: {2,2,4}     91: {4,6}
     29: {10}        66: {1,2,5}     94: {1,15}
     30: {1,2,3}     70: {1,3,4}    100: {1,1,3,3}
     34: {1,7}       71: {20}       101: {26}
For example, 75 has 3 prime indices {2,3,3} with sum 8 >= 2*3, so 75 is in the sequence.

Maple

filter:= proc(n) local F, a, t;
  F:= ifactors(n)[2];
  a:= add((numtheory:-pi(t[1])-2)*t[2], t=F);
  a::even and a >= 0
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 10 2024

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 A209816 and A110618.

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

The conjugate version appears to be A320924 (allowing odd weights: A322109).

The case of equality is A340387.

Allowing odd weights gives A344291.

The 5-smooth case is A344295, or A344293 allowing odd weights.

The opposite version allowing odd weights is A344296.

The conjugate opposite version allowing odd weights is A344414.

The case of equality in the conjugate case is A344415.

The conjugate opposite 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, A025065, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.

Sequence in context: A099248 A082575 A284526 * A339618 A357707 A294571

Adjacent sequences: A344410 A344411 A344412 * A344414 A344415 A344416

Keyword

nonn

Author

Gus Wiseman, May 19 2021

Status

approved