A328603 Numbers whose prime indices have no consecutive divisible parts, meaning no prime index is a divisor of the next-smallest prime index, counted with multiplicity.

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167

Offset: 1

Author

Gus Wiseman, Oct 26 2019

Keywords

• nonn

Comments

First differs from A304713 in having 105, with prime indices {2, 3, 4}.

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 sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   31: {11}
   33: {2,5}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   51: {2,7}
		

Crossrefs

A subset of A005117.

These are the Heinz numbers of the partitions counted by A328171.

The non-strict version is A328674 (Heinz numbers for A328675).

The version for relatively prime instead of indivisible is A328335.

Compositions without consecutive divisibilities are A328460.

Numbers whose binary indices lack consecutive divisibilities are A328593.

The version with all pairs indivisible is A304713.

Cf. A056239, A112798, A316476, A318726, A318729, A326704, A328336, A328598, A328599.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MatchQ[primeMS[#],{_,x_,y_,_}/;Divisible[y,x]]&]

Formula

Intersection of A005117 and A328674.