A367401 Numbers m such that bigomega(m) is not the sum of prime indices of any semiprime divisor of m.

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Author

Gus Wiseman, Nov 21 2023

Keywords

• nonn

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 the partitions counted by A367398.

Examples

			60 has semiprime divisor 10 with prime indices {1,3} summing to 4 = bigomega(60), so 60 is not in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
		

Crossrefs

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

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

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

partitions: A367212 A367213 A367218 A367219 A367394 A367398

strict: A367214 A367215 A367220 A367221 A367395 A367399

subsets: A367216 A367217 A367222 A367223 A367396 A367400

ranks: A367224 A367225 A367226 A367227 A367397 A367401*

A002865 counts partitions w/ length, complement A229816, ranks A325761.

A088809 and A093971 count subsets containing semi-sums.

A236912 counts partitions with no semi-sum of the parts, ranks A364461.

A237113 counts partitions with a semi-sum of the parts, ranks A364462.

A366738 counts semi-sums of partitions, strict A366741.

Triangles:

A365381 counts subsets with a subset summing to k, complement A366320.

A365541 counts subsets with a semi-sum k.

A367404 counts partitions with a semi-sum k, strict A367405.

Cf. A000700, A237667, A238628, A304792, A363225, A364272, A365543, A365658, A365918, A366740.

Programs

Mathematica

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