A368833 Numbers whose prime indices are not 1, prime, or semiprime.

19, 37, 38, 53, 57, 61, 71, 74, 76, 89, 95, 103, 106, 107, 111, 113, 114, 122, 131, 133, 142, 148, 151, 152, 159, 171, 173, 178, 181, 183, 185, 190, 193, 197, 206, 209, 212, 213, 214, 222, 223, 226, 228, 229, 239, 244, 247, 251, 259, 262, 263, 265, 266, 267

Offset: 1

Author

Gus Wiseman, Jan 08 2024

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.

Examples

			The terms together with their prime indices begin:
   19: {8}
   37: {12}
   38: {1,8}
   53: {16}
   57: {2,8}
   61: {18}
   71: {20}
   74: {1,12}
   76: {1,1,8}
   89: {24}
   95: {3,8}
  103: {27}
  106: {1,16}
  107: {28}
  111: {2,12}
  113: {30}
  114: {1,2,8}
  122: {1,18}
  131: {32}
  133: {4,8}
  142: {1,20}
  148: {1,1,12}
		

Crossrefs

These are non-products of primes indexed by elements of A037143.

The complement for just primes is A076610, strict A302590.

The complement for just semiprimes is A339112, strict A340020.

The complement for just squarefree semiprimes is A339113, strict A309356.

The complement is A368728.

The complement for just primes and semiprimes is A368729, strict A340019.

A000607 counts partitions into primes, with ones allowed A034891.

A001358 lists semiprimes, squarefree A006881.

A006450, A106349, A322551, A368732 list selected primes.

A056239 adds up prime indices, row sums of A112798.

A101048 counts partitions into semiprimes.

Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628.

Programs

Mathematica

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