A328867 Heinz numbers of integer partitions in which no two distinct parts are relatively prime.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 64, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 128, 129, 131, 133, 137, 139, 147, 149

Offset: 1

Author

Gus Wiseman, Oct 30 2019

Keywords

• nonn

Comments

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 relatively prime is said to be intersecting.

Examples

			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}
   21: {2,4}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
		

Crossrefs

These are the Heinz numbers of the partitions counted by A328673.

The strict case is A318719.

The relatively prime version is A328868.

A ranking using binary indices is A326910.

The version for non-isomorphic multiset partitions is A319752.

The version for divisibility (instead of relative primality) is A316476.

Cf. A000837, A056239, A112798, A200976, A289509, A303283, A305843, A318715, A318716, A328336.

Programs

Mathematica

Select[Range[100],And@@(GCD[##]>1&)@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]