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

1,2

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.

Links

Table of n, a(n) for n=1..60.

Formula

Intersection of A005117 and A328674.

Example

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}

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]]&]

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.

Sequence in context: A072697 A102553 A069161 * A304713 A319327 A319319

Adjacent sequences: A328600 A328601 A328602 * A328604 A328605 A328606

Keyword

nonn

Author

Gus Wiseman, Oct 26 2019

Status

approved