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

1,2

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.

Links

Table of n, a(n) for n=1..62.

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

Mathematica

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

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.

Sequence in context: A360117 A014567 A324769 * A326536 A322902 A302040

Adjacent sequences: A328864 A328865 A328866 * A328868 A328869 A328870

Keyword

nonn

Author

Gus Wiseman, Oct 30 2019

Status

approved