A367224 Numbers m with a divisor whose prime indices sum to bigomega(m).

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 30, 32, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 56, 57, 60, 64, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 90, 93, 96, 100, 102, 105, 108, 110, 111, 112, 114, 120, 123, 125, 126, 128, 129, 130, 132, 135, 138, 140, 141

Offset

1,2

Comments

Also numbers m whose prime indices have a submultiset summing to bigomega(m).

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 the partitions counted by A367212.

Links

Table of n, a(n) for n=1..62.

Example

The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}

Mathematica

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

Crossrefs

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.

sum-full sum-free comb-full comb-free

-------------------------------------------

partitions: A367212 A367213 A367218 A367219

strict: A367214 A367215 A367220 A367221

subsets: A367216 A367217 A367222 A367223

ranks: A367224* A367225 A367226 A367227

A000700 counts self-conjugate partitions, ranks A088902.

A002865 counts partitions whose length is a part, ranks A325761.

A005117 ranks strict integer partitions, counted by A000009.

A066208 ranks partitions into odd parts, also counted by A000009.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A126796 counts complete partitions, ranks A325781.

A229816 counts partitions whose length is not a part, ranks A367107.

A237668 counts sum-full partitions, ranks A364532.

Triangles:

A046663 counts partitions of n without a subset-sum k, strict A365663.

A365543 counts partitions of n with a subset-sum k, strict A365661.

A365658 counts partitions by number of subset-sums, strict A365832.

Cf. A000720, A055396, A061395, A106529, A299729, A304792, A363225, A365830.

Sequence in context: A364123 A244799 A347262 * A316350 A335127 A071562

Adjacent sequences: A367221 A367222 A367223 * A367225 A367226 A367227

Keyword

nonn

Author

Gus Wiseman, Nov 14 2023

Status

approved