A329366 Numbers whose distinct prime indices are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197

Offset

1,2

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are Heinz numbers of stable intersecting partitions.

Links

Table of n, a(n) for n=1..61.

Example

The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[100], stableQ[Union[primeMS[#]], GCD[#1, #2]==1&]&&stableQ[Union[primeMS[#]], Divisible]&]

Crossrefs

Intersection of A316476 and A328867.

Heinz numbers of the partitions counted by A328871.

Replacing "intersecting" with "relatively prime" gives A328677.

Cf. A056239, A112798, A285573, A289509, A303362, A304713, A327393, A328671.

Sequence in context: A325371 A059046 A363727 * A144711 A337935 A036116

Adjacent sequences: A329363 A329364 A329365 * A329367 A329368 A329369

Keyword

nonn

Author

Gus Wiseman, Nov 12 2019

Status

approved